sdclem2 - Mathbox for Jeff Madsen

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Theorem sdclem2 32708

Description: Lemma for sdc 32710. (Contributed by Jeff Madsen, 2-Sep-2009.)

**Hypotheses**
Ref	Expression
sdc.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
sdc.2	⊢ (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓 ↔ 𝜒))
sdc.3	⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜏))
sdc.4	⊢ (𝑛 = 𝑘 → (𝜓 ↔ 𝜃))
sdc.5	⊢ ((𝑔 = ℎ ∧ 𝑛 = (𝑘 + 1)) → (𝜓 ↔ 𝜎))
sdc.6	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sdc.7	⊢ (𝜑 → 𝑀 ∈ ℤ)
sdc.8	⊢ (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴 ∧ 𝜏))
sdc.9	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃) → ∃ℎ(ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑔 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)))
sdc.10	⊢ 𝐽 = {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)}
sdc.11	⊢ 𝐹 = (𝑤 ∈ 𝑍, 𝑥 ∈ 𝐽 ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})
sdc.12	⊢ Ⅎ𝑘𝜑
sdc.13	⊢ (𝜑 → 𝐺:𝑍⟶𝐽)
sdc.14	⊢ (𝜑 → (𝐺‘𝑀):(𝑀...𝑀)⟶𝐴)
sdc.15	⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝐺‘(𝑤 + 1)) ∈ (𝑤𝐹(𝐺‘𝑤)))

**Assertion**
Ref	Expression
sdclem2	⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒))

Distinct variable groups: 𝑓,𝑔,ℎ,𝑘,𝑛,𝑤,𝑥,𝐴 ℎ,𝐽,𝑘,𝑤,𝑥 𝑓,𝑀,𝑔,ℎ,𝑘,𝑛,𝑤,𝑥 𝜒,𝑔 𝑛,𝐹,𝑤,𝑥 𝜓,𝑓,ℎ,𝑘,𝑥 𝜎,𝑓,𝑔,𝑛,𝑥 𝑓,𝐺,𝑔,ℎ,𝑘,𝑛,𝑤,𝑥 𝜑,𝑛,𝑤,𝑥 𝜃,𝑛,𝑤,𝑥 ℎ,𝑉 𝜏,ℎ,𝑘,𝑛,𝑤,𝑥 𝑓,𝑍,𝑔,ℎ,𝑘,𝑛,𝑤,𝑥 Allowed substitution hints: 𝜑(𝑓,𝑔,ℎ,𝑘) 𝜓(𝑤,𝑔,𝑛) 𝜒(𝑥,𝑤,𝑓,ℎ,𝑘,𝑛) 𝜃(𝑓,𝑔,ℎ,𝑘) 𝜏(𝑓,𝑔) 𝜎(𝑤,ℎ,𝑘) 𝐹(𝑓,𝑔,ℎ,𝑘) 𝐽(𝑓,𝑔,𝑛) 𝑉(𝑥,𝑤,𝑓,𝑔,𝑘,𝑛)

Proof of Theorem sdclem2 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sdc.12	. . 3 ⊢ Ⅎ𝑘𝜑
2		sdc.13	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑍⟶𝐽)
3	2	ffvelrnda 6267	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐽)
4		sdc.10	. . . . . . . . 9 ⊢ 𝐽 = {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)}
5	4	eleq2i 2680	. . . . . . . 8 ⊢ ((𝐺‘𝑘) ∈ 𝐽 ↔ (𝐺‘𝑘) ∈ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)})
6		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑔𝑍
7		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑔(𝐺‘𝑘):(𝑀...𝑛)⟶𝐴
8		nfsbc1v 3422	. . . . . . . . . . 11 ⊢ Ⅎ𝑔[(𝐺‘𝑘) / 𝑔]𝜓
9	7, 8	nfan 1816	. . . . . . . . . 10 ⊢ Ⅎ𝑔((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓)
10	6, 9	nfrex 2990	. . . . . . . . 9 ⊢ Ⅎ𝑔∃𝑛 ∈ 𝑍 ((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓)
11		fvex 6113	. . . . . . . . 9 ⊢ (𝐺‘𝑘) ∈ V
12		feq1 5939	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐺‘𝑘) → (𝑔:(𝑀...𝑛)⟶𝐴 ↔ (𝐺‘𝑘):(𝑀...𝑛)⟶𝐴))
13		sbceq1a 3413	. . . . . . . . . . 11 ⊢ (𝑔 = (𝐺‘𝑘) → (𝜓 ↔ [(𝐺‘𝑘) / 𝑔]𝜓))
14	12, 13	anbi12d 743	. . . . . . . . . 10 ⊢ (𝑔 = (𝐺‘𝑘) → ((𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓) ↔ ((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓)))
15	14	rexbidv 3034	. . . . . . . . 9 ⊢ (𝑔 = (𝐺‘𝑘) → (∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓) ↔ ∃𝑛 ∈ 𝑍 ((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓)))
16	10, 11, 15	elabf 3318	. . . . . . . 8 ⊢ ((𝐺‘𝑘) ∈ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} ↔ ∃𝑛 ∈ 𝑍 ((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓))
17	5, 16	bitri 263	. . . . . . 7 ⊢ ((𝐺‘𝑘) ∈ 𝐽 ↔ ∃𝑛 ∈ 𝑍 ((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓))
18	3, 17	sylib 207	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∃𝑛 ∈ 𝑍 ((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓))
19		fdm 5964	. . . . . . . . . 10 ⊢ ((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 → dom (𝐺‘𝑘) = (𝑀...𝑛))
20	19	adantr 480	. . . . . . . . 9 ⊢ (((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓) → dom (𝐺‘𝑘) = (𝑀...𝑛))
21		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑀 → (𝐺‘𝑥) = (𝐺‘𝑀))
22		oveq2 6557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
23	22	mpteq1d 4666	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑀 → (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚)))
24	21, 23	eqeq12d 2625	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑀 → ((𝐺‘𝑥) = (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚)) ↔ (𝐺‘𝑀) = (𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚))))
25	24	imbi2d 329	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑀 → ((𝜑 → (𝐺‘𝑥) = (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚))) ↔ (𝜑 → (𝐺‘𝑀) = (𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚)))))
26		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝐺‘𝑥) = (𝐺‘𝑤))
27		oveq2 6557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑤 → (𝑀...𝑥) = (𝑀...𝑤))
28	27	mpteq1d 4666	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))
29	26, 28	eqeq12d 2625	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑤 → ((𝐺‘𝑥) = (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚)) ↔ (𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚))))
30	29	imbi2d 329	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → ((𝜑 → (𝐺‘𝑥) = (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚))) ↔ (𝜑 → (𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))))
31		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑤 + 1) → (𝐺‘𝑥) = (𝐺‘(𝑤 + 1)))
32		oveq2 6557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑤 + 1) → (𝑀...𝑥) = (𝑀...(𝑤 + 1)))
33	32	mpteq1d 4666	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑤 + 1) → (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)))
34	31, 33	eqeq12d 2625	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑤 + 1) → ((𝐺‘𝑥) = (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚)) ↔ (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))))
35	34	imbi2d 329	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑤 + 1) → ((𝜑 → (𝐺‘𝑥) = (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚))) ↔ (𝜑 → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)))))
36		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → (𝐺‘𝑥) = (𝐺‘𝑘))
37		oveq2 6557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑘 → (𝑀...𝑥) = (𝑀...𝑘))
38	37	mpteq1d 4666	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)))
39	36, 38	eqeq12d 2625	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑘 → ((𝐺‘𝑥) = (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚)) ↔ (𝐺‘𝑘) = (𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚))))
40	39	imbi2d 329	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → ((𝜑 → (𝐺‘𝑥) = (𝑚 ∈ (𝑀...𝑥) ↦ ((𝐺‘𝑚)‘𝑚))) ↔ (𝜑 → (𝐺‘𝑘) = (𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)))))
41		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 = 𝑘 → (𝐺‘𝑚) = (𝐺‘𝑘))
42		id 22	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 = 𝑘 → 𝑚 = 𝑘)
43	41, 42	fveq12d 6109	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = 𝑘 → ((𝐺‘𝑚)‘𝑚) = ((𝐺‘𝑘)‘𝑘))
44		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚))
45		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺‘𝑘)‘𝑘) ∈ V
46	43, 44, 45	fvmpt 6191	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (𝑀...𝑀) → ((𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑘) = ((𝐺‘𝑘)‘𝑘))
47	46	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑀)) → ((𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑘) = ((𝐺‘𝑘)‘𝑘))
48		elfz1eq 12223	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (𝑀...𝑀) → 𝑘 = 𝑀)
49	48	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑀)) → 𝑘 = 𝑀)
50	49	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑀)) → (𝐺‘𝑘) = (𝐺‘𝑀))
51	50	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑀)) → ((𝐺‘𝑘)‘𝑘) = ((𝐺‘𝑀)‘𝑘))
52	47, 51	eqtr2d 2645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑀)) → ((𝐺‘𝑀)‘𝑘) = ((𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑘))
53	52	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑘 ∈ (𝑀...𝑀) → ((𝐺‘𝑀)‘𝑘) = ((𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑘)))
54	1, 53	ralrimi 2940	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑀)((𝐺‘𝑀)‘𝑘) = ((𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑘))
55		sdc.14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐺‘𝑀):(𝑀...𝑀)⟶𝐴)
56		ffn 5958	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺‘𝑀):(𝑀...𝑀)⟶𝐴 → (𝐺‘𝑀) Fn (𝑀...𝑀))
57	55, 56	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺‘𝑀) Fn (𝑀...𝑀))
58		fvex 6113	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺‘𝑚)‘𝑚) ∈ V
59	58, 44	fnmpti 5935	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚)) Fn (𝑀...𝑀)
60		eqfnfv 6219	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺‘𝑀) Fn (𝑀...𝑀) ∧ (𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚)) Fn (𝑀...𝑀)) → ((𝐺‘𝑀) = (𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚)) ↔ ∀𝑘 ∈ (𝑀...𝑀)((𝐺‘𝑀)‘𝑘) = ((𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑘)))
61	57, 59, 60	sylancl 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐺‘𝑀) = (𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚)) ↔ ∀𝑘 ∈ (𝑀...𝑀)((𝐺‘𝑀)‘𝑘) = ((𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑘)))
62	54, 61	mpbird 246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺‘𝑀) = (𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚)))
63	62	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝐺‘𝑀) = (𝑚 ∈ (𝑀...𝑀) ↦ ((𝐺‘𝑚)‘𝑚))))
64		sdc.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑍 = (ℤ_≥‘𝑀)
65	64	eleq2i 2680	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ 𝑍 ↔ 𝑤 ∈ (ℤ_≥‘𝑀))
66	2	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝐺‘𝑤) ∈ 𝐽)
67		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍)
68		3simpa 1051	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎) → (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘))))
69	68	reximi 2994	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎) → ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘))))
70	69	ss2abi 3637	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ⊆ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))}
71		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ℤ_≥‘𝑀) ∈ V
72	64, 71	eqeltri 2684	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑍 ∈ V
73		nfv 1830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ Ⅎ𝑘 𝑤 ∈ 𝑍
74	1, 73	nfan 1816	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑤 ∈ 𝑍)
75		sdc.6	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
76	75	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝐴 ∈ 𝑉)
77		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘))) → ℎ:(𝑀...(𝑘 + 1))⟶𝐴)
78		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑀...(𝑘 + 1)) ∈ V
79		elmapg 7757	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑀...(𝑘 + 1)) ∈ V) → (ℎ ∈ (𝐴 ↑_𝑚 (𝑀...(𝑘 + 1))) ↔ ℎ:(𝑀...(𝑘 + 1))⟶𝐴))
80	78, 79	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝐴 ∈ 𝑉 → (ℎ ∈ (𝐴 ↑_𝑚 (𝑀...(𝑘 + 1))) ↔ ℎ:(𝑀...(𝑘 + 1))⟶𝐴))
81	77, 80	syl5ibr 235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐴 ∈ 𝑉 → ((ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘))) → ℎ ∈ (𝐴 ↑_𝑚 (𝑀...(𝑘 + 1)))))
82	81	abssdv 3639	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ⊆ (𝐴 ↑_𝑚 (𝑀...(𝑘 + 1))))
83	76, 82	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {ℎ ∣ (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ⊆ (𝐴 ↑_𝑚 (𝑀...(𝑘 + 1))))
84		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐴 ↑_𝑚 (𝑀...(𝑘 + 1))) ∈ V
85		ssexg 4732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (({ℎ ∣ (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ⊆ (𝐴 ↑_𝑚 (𝑀...(𝑘 + 1))) ∧ (𝐴 ↑_𝑚 (𝑀...(𝑘 + 1))) ∈ V) → {ℎ ∣ (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ∈ V)
86	83, 84, 85	sylancl 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {ℎ ∣ (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ∈ V)
87	86	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑘 ∈ 𝑍 → {ℎ ∣ (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ∈ V))
88	74, 87	ralrimi 2940	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ∀𝑘 ∈ 𝑍 {ℎ ∣ (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ∈ V)
89		abrexex2g 7036	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑍 ∈ V ∧ ∀𝑘 ∈ 𝑍 {ℎ ∣ (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ∈ V) → {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ∈ V)
90	72, 88, 89	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ∈ V)
91		ssexg 4732	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (({ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ⊆ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ∧ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ∈ V) → {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V)
92	70, 90, 91	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V)
93		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = (𝐺‘𝑤) → (𝑥 = (ℎ ↾ (𝑀...𝑘)) ↔ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘))))
94	93	3anbi2d 1396	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = (𝐺‘𝑤) → ((ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)))
95	94	rexbidv 3034	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = (𝐺‘𝑤) → (∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)))
96	95	abbidv 2728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = (𝐺‘𝑤) → {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})
97	96	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = (𝐺‘𝑤) → ({ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V ↔ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V))
98		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = (𝐺‘𝑤) → (𝑤𝐹𝑥) = (𝑤𝐹(𝐺‘𝑤)))
99	98, 96	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = (𝐺‘𝑤) → ((𝑤𝐹𝑥) = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ↔ (𝑤𝐹(𝐺‘𝑤)) = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)}))
100	97, 99	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = (𝐺‘𝑤) → (({ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V → (𝑤𝐹𝑥) = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)}) ↔ ({ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V → (𝑤𝐹(𝐺‘𝑤)) = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})))
101	100	imbi2d 329	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = (𝐺‘𝑤) → ((𝑤 ∈ 𝑍 → ({ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V → (𝑤𝐹𝑥) = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})) ↔ (𝑤 ∈ 𝑍 → ({ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V → (𝑤𝐹(𝐺‘𝑤)) = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)}))))
102		sdc.11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝐹 = (𝑤 ∈ 𝑍, 𝑥 ∈ 𝐽 ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})
103	102	ovmpt4g 6681	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑤 ∈ 𝑍 ∧ 𝑥 ∈ 𝐽 ∧ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V) → (𝑤𝐹𝑥) = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})
104	103	3com12 1261	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑤 ∈ 𝑍 ∧ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V) → (𝑤𝐹𝑥) = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})
105	104	3exp 1256	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ 𝐽 → (𝑤 ∈ 𝑍 → ({ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V → (𝑤𝐹𝑥) = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})))
106	101, 105	vtoclga 3245	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺‘𝑤) ∈ 𝐽 → (𝑤 ∈ 𝑍 → ({ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} ∈ V → (𝑤𝐹(𝐺‘𝑤)) = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})))
107	66, 67, 92, 106	syl3c 64	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤𝐹(𝐺‘𝑤)) = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})
108	107, 70	syl6eqss 3618	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤𝐹(𝐺‘𝑤)) ⊆ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))})
109		sdc.15	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝐺‘(𝑤 + 1)) ∈ (𝑤𝐹(𝐺‘𝑤)))
110	108, 109	sseldd 3569	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝐺‘(𝑤 + 1)) ∈ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))})
111		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐺‘(𝑤 + 1)) ∈ V
112		feq1 5939	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ = (𝐺‘(𝑤 + 1)) → (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ↔ (𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴))
113		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ = (𝐺‘(𝑤 + 1)) → (ℎ ↾ (𝑀...𝑘)) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)))
114	113	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ = (𝐺‘(𝑤 + 1)) → ((𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)) ↔ (𝐺‘𝑤) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘))))
115	112, 114	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = (𝐺‘(𝑤 + 1)) → ((ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘))) ↔ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)))))
116	115	rexbidv 3034	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = (𝐺‘(𝑤 + 1)) → (∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘))) ↔ ∃𝑘 ∈ 𝑍 ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)))))
117	111, 116	elab 3319	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺‘(𝑤 + 1)) ∈ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = (ℎ ↾ (𝑀...𝑘)))} ↔ ∃𝑘 ∈ 𝑍 ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘))))
118	110, 117	sylib 207	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ∃𝑘 ∈ 𝑍 ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘))))
119		nfv 1830	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘((𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)))
120		simprl 790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴)
121		fzssp1 12255	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑀...𝑘) ⊆ (𝑀...(𝑘 + 1))
122		fssres 5983	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝑀...𝑘) ⊆ (𝑀...(𝑘 + 1))) → ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)):(𝑀...𝑘)⟶𝐴)
123	120, 121, 122	sylancl 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)):(𝑀...𝑘)⟶𝐴)
124		fdm 5964	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)):(𝑀...𝑘)⟶𝐴 → dom ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑀...𝑘))
125	123, 124	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → dom ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑀...𝑘))
126		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚))
127	58, 126	fnmpti 5935	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) Fn (𝑀...𝑤)
128		simprr 792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))
129	128	fneq1d 5895	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) Fn (𝑀...𝑤) ↔ (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) Fn (𝑀...𝑤)))
130	127, 129	mpbiri 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) Fn (𝑀...𝑤))
131		fndm 5904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) Fn (𝑀...𝑤) → dom ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑀...𝑤))
132	130, 131	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → dom ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑀...𝑤))
133	125, 132	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝑀...𝑘) = (𝑀...𝑤))
134		simplr 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → 𝑘 ∈ 𝑍)
135	134, 64	syl6eleq 2698	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → 𝑘 ∈ (ℤ_≥‘𝑀))
136		fzopth 12249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → ((𝑀...𝑘) = (𝑀...𝑤) ↔ (𝑀 = 𝑀 ∧ 𝑘 = 𝑤)))
137	135, 136	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → ((𝑀...𝑘) = (𝑀...𝑤) ↔ (𝑀 = 𝑀 ∧ 𝑘 = 𝑤)))
138	133, 137	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝑀 = 𝑀 ∧ 𝑘 = 𝑤))
139	138	simprd 478	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → 𝑘 = 𝑤)
140	139	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝑘 + 1) = (𝑤 + 1))
141	140	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝑀...(𝑘 + 1)) = (𝑀...(𝑤 + 1)))
142		elfzp1 12261	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑥 ∈ (𝑀...(𝑘 + 1)) ↔ (𝑥 ∈ (𝑀...𝑘) ∨ 𝑥 = (𝑘 + 1))))
143	135, 142	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝑥 ∈ (𝑀...(𝑘 + 1)) ↔ (𝑥 ∈ (𝑀...𝑘) ∨ 𝑥 = (𝑘 + 1))))
144	133	reseq2d 5317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑘)) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑤)))
145		fzssp1 12255	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑀...𝑤) ⊆ (𝑀...(𝑤 + 1))
146		resmpt 5369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑀...𝑤) ⊆ (𝑀...(𝑤 + 1)) → ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑤)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))
147	145, 146	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑤)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚))
148	144, 147	syl6req 2661	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑘)))
149	128, 148	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑘)))
150	149	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘))‘𝑥) = (((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑘))‘𝑥))
151		fvres 6117	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 ∈ (𝑀...𝑘) → (((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘))‘𝑥) = ((𝐺‘(𝑤 + 1))‘𝑥))
152		fvres 6117	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 ∈ (𝑀...𝑘) → (((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑘))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥))
153	151, 152	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 ∈ (𝑀...𝑘) → ((((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘))‘𝑥) = (((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑘))‘𝑥) ↔ ((𝐺‘(𝑤 + 1))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥)))
154	150, 153	syl5ibcom 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝑥 ∈ (𝑀...𝑘) → ((𝐺‘(𝑤 + 1))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥)))
155	140	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝑥 = (𝑘 + 1) ↔ 𝑥 = (𝑤 + 1)))
156	139, 135	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → 𝑤 ∈ (ℤ_≥‘𝑀))
157		peano2uz 11617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑤 ∈ (ℤ_≥‘𝑀) → (𝑤 + 1) ∈ (ℤ_≥‘𝑀))
158		eluzfz2 12220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑤 + 1) ∈ (ℤ_≥‘𝑀) → (𝑤 + 1) ∈ (𝑀...(𝑤 + 1)))
159		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑚 = (𝑤 + 1) → (𝐺‘𝑚) = (𝐺‘(𝑤 + 1)))
160		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑚 = (𝑤 + 1) → 𝑚 = (𝑤 + 1))
161	159, 160	fveq12d 6109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑚 = (𝑤 + 1) → ((𝐺‘𝑚)‘𝑚) = ((𝐺‘(𝑤 + 1))‘(𝑤 + 1)))
162		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))
163		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐺‘(𝑤 + 1))‘(𝑤 + 1)) ∈ V
164	161, 162, 163	fvmpt 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑤 + 1) ∈ (𝑀...(𝑤 + 1)) → ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘(𝑤 + 1)) = ((𝐺‘(𝑤 + 1))‘(𝑤 + 1)))
165	156, 157, 158, 164	4syl 19	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘(𝑤 + 1)) = ((𝐺‘(𝑤 + 1))‘(𝑤 + 1)))
166	165	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → ((𝐺‘(𝑤 + 1))‘(𝑤 + 1)) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘(𝑤 + 1)))
167		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑥 = (𝑤 + 1) → ((𝐺‘(𝑤 + 1))‘𝑥) = ((𝐺‘(𝑤 + 1))‘(𝑤 + 1)))
168		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑥 = (𝑤 + 1) → ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘(𝑤 + 1)))
169	167, 168	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = (𝑤 + 1) → (((𝐺‘(𝑤 + 1))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥) ↔ ((𝐺‘(𝑤 + 1))‘(𝑤 + 1)) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘(𝑤 + 1))))
170	166, 169	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝑥 = (𝑤 + 1) → ((𝐺‘(𝑤 + 1))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥)))
171	155, 170	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝑥 = (𝑘 + 1) → ((𝐺‘(𝑤 + 1))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥)))
172	154, 171	jaod 394	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → ((𝑥 ∈ (𝑀...𝑘) ∨ 𝑥 = (𝑘 + 1)) → ((𝐺‘(𝑤 + 1))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥)))
173	143, 172	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝑥 ∈ (𝑀...(𝑘 + 1)) → ((𝐺‘(𝑤 + 1))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥)))
174	173	ralrimiv 2948	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → ∀𝑥 ∈ (𝑀...(𝑘 + 1))((𝐺‘(𝑤 + 1))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥))
175		ffn 5958	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 → (𝐺‘(𝑤 + 1)) Fn (𝑀...(𝑘 + 1)))
176	175	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝐺‘(𝑤 + 1)) Fn (𝑀...(𝑘 + 1)))
177	58, 162	fnmpti 5935	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) Fn (𝑀...(𝑤 + 1))
178		eqfnfv2 6220	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐺‘(𝑤 + 1)) Fn (𝑀...(𝑘 + 1)) ∧ (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) Fn (𝑀...(𝑤 + 1))) → ((𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) ↔ ((𝑀...(𝑘 + 1)) = (𝑀...(𝑤 + 1)) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))((𝐺‘(𝑤 + 1))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥))))
179	176, 177, 178	sylancl 693	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → ((𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)) ↔ ((𝑀...(𝑘 + 1)) = (𝑀...(𝑤 + 1)) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))((𝐺‘(𝑤 + 1))‘𝑥) = ((𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))‘𝑥))))
180	141, 174, 179	mpbir2and 959	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)))) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)))
181	180	expr 641	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ (𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴) → (((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))))
182		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐺‘𝑤) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) → ((𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) ↔ ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚))))
183	182	imbi1d 330	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺‘𝑤) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) → (((𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))) ↔ (((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)))))
184	181, 183	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) ∧ (𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴) → ((𝐺‘𝑤) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘)) → ((𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)))))
185	184	expimpd 627	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘))) → ((𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)))))
186	185	ex 449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑘 ∈ 𝑍 → (((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘))) → ((𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))))))
187	74, 119, 186	rexlimd 3008	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (∃𝑘 ∈ 𝑍 ((𝐺‘(𝑤 + 1)):(𝑀...(𝑘 + 1))⟶𝐴 ∧ (𝐺‘𝑤) = ((𝐺‘(𝑤 + 1)) ↾ (𝑀...𝑘))) → ((𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)))))
188	118, 187	mpd 15	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ((𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚))))
189	188	expcom 450	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ 𝑍 → (𝜑 → ((𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)))))
190	65, 189	sylbir 224	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (ℤ_≥‘𝑀) → (𝜑 → ((𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚)) → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)))))
191	190	a2d 29	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (ℤ_≥‘𝑀) → ((𝜑 → (𝐺‘𝑤) = (𝑚 ∈ (𝑀...𝑤) ↦ ((𝐺‘𝑚)‘𝑚))) → (𝜑 → (𝐺‘(𝑤 + 1)) = (𝑚 ∈ (𝑀...(𝑤 + 1)) ↦ ((𝐺‘𝑚)‘𝑚)))))
192	25, 30, 35, 40, 63, 191	uzind4 11622	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝜑 → (𝐺‘𝑘) = (𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚))))
193	192, 64	eleq2s 2706	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → (𝐺‘𝑘) = (𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚))))
194	193	impcom 445	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)))
195	194	dmeqd 5248	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝐺‘𝑘) = dom (𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)))
196		dmmptg 5549	. . . . . . . . . . . . . 14 ⊢ (∀𝑚 ∈ (𝑀...𝑘)((𝐺‘𝑚)‘𝑚) ∈ V → dom (𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑀...𝑘))
197	58	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝑀...𝑘) → ((𝐺‘𝑚)‘𝑚) ∈ V)
198	196, 197	mprg 2910	. . . . . . . . . . . . 13 ⊢ dom (𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑀...𝑘)
199	195, 198	syl6eq 2660	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝐺‘𝑘) = (𝑀...𝑘))
200	199	eqeq1d 2612	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (dom (𝐺‘𝑘) = (𝑀...𝑛) ↔ (𝑀...𝑘) = (𝑀...𝑛)))
201		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
202	201, 64	syl6eleq 2698	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (ℤ_≥‘𝑀))
203		fzopth 12249	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → ((𝑀...𝑘) = (𝑀...𝑛) ↔ (𝑀 = 𝑀 ∧ 𝑘 = 𝑛)))
204	202, 203	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑀...𝑘) = (𝑀...𝑛) ↔ (𝑀 = 𝑀 ∧ 𝑘 = 𝑛)))
205	200, 204	bitrd 267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (dom (𝐺‘𝑘) = (𝑀...𝑛) ↔ (𝑀 = 𝑀 ∧ 𝑘 = 𝑛)))
206		simpr 476	. . . . . . . . . 10 ⊢ ((𝑀 = 𝑀 ∧ 𝑘 = 𝑛) → 𝑘 = 𝑛)
207	205, 206	syl6bi 242	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (dom (𝐺‘𝑘) = (𝑀...𝑛) → 𝑘 = 𝑛))
208	20, 207	syl5 33	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓) → 𝑘 = 𝑛))
209		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
210	209	feq2d 5944	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ↔ (𝐺‘𝑘):(𝑀...𝑘)⟶𝐴))
211		sdc.4	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝜓 ↔ 𝜃))
212	211	sbcbidv 3457	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ([(𝐺‘𝑘) / 𝑔]𝜓 ↔ [(𝐺‘𝑘) / 𝑔]𝜃))
213	210, 212	anbi12d 743	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓) ↔ ((𝐺‘𝑘):(𝑀...𝑘)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜃)))
214	213	equcoms 1934	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓) ↔ ((𝐺‘𝑘):(𝑀...𝑘)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜃)))
215	214	biimpcd 238	. . . . . . . 8 ⊢ (((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓) → (𝑘 = 𝑛 → ((𝐺‘𝑘):(𝑀...𝑘)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜃)))
216	208, 215	sylcom 30	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓) → ((𝐺‘𝑘):(𝑀...𝑘)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜃)))
217	216	rexlimdvw 3016	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (∃𝑛 ∈ 𝑍 ((𝐺‘𝑘):(𝑀...𝑛)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜓) → ((𝐺‘𝑘):(𝑀...𝑘)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜃)))
218	18, 217	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑘):(𝑀...𝑘)⟶𝐴 ∧ [(𝐺‘𝑘) / 𝑔]𝜃))
219	218	simpld 474	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘):(𝑀...𝑘)⟶𝐴)
220		eluzfz2 12220	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ (𝑀...𝑘))
221	202, 220	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (𝑀...𝑘))
222	219, 221	ffvelrnd 6268	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑘)‘𝑘) ∈ 𝐴)
223	43	cbvmptv 4678	. . 3 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑘 ∈ 𝑍 ↦ ((𝐺‘𝑘)‘𝑘))
224	1, 222, 223	fmptdf 6294	. 2 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)):𝑍⟶𝐴)
225	218	simprd 478	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → [(𝐺‘𝑘) / 𝑔]𝜃)
226	194, 225	sbceq1dd 3408	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → [(𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜃)
227	226	ex 449	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 → [(𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜃))
228	1, 227	ralrimi 2940	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 [(𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜃)
229		mpteq1 4665	. . . . . 6 ⊢ ((𝑀...𝑛) = (𝑀...𝑘) → (𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)))
230		dfsbcq 3404	. . . . . 6 ⊢ ((𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) = (𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) → ([(𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓 ↔ [(𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓))
231	209, 229, 230	3syl 18	. . . . 5 ⊢ (𝑛 = 𝑘 → ([(𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓 ↔ [(𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓))
232	211	sbcbidv 3457	. . . . 5 ⊢ (𝑛 = 𝑘 → ([(𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓 ↔ [(𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜃))
233	231, 232	bitrd 267	. . . 4 ⊢ (𝑛 = 𝑘 → ([(𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓 ↔ [(𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜃))
234	233	cbvralv 3147	. . 3 ⊢ (∀𝑛 ∈ 𝑍 [(𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓 ↔ ∀𝑘 ∈ 𝑍 [(𝑚 ∈ (𝑀...𝑘) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜃)
235	228, 234	sylibr 223	. 2 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 [(𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓)
236	72	mptex 6390	. . 3 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) ∈ V
237		feq1 5939	. . . 4 ⊢ (𝑓 = (𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) → (𝑓:𝑍⟶𝐴 ↔ (𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)):𝑍⟶𝐴))
238		vex 3176	. . . . . . . 8 ⊢ 𝑓 ∈ V
239	238	resex 5363	. . . . . . 7 ⊢ (𝑓 ↾ (𝑀...𝑛)) ∈ V
240		sdc.2	. . . . . . 7 ⊢ (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓 ↔ 𝜒))
241	239, 240	sbcie 3437	. . . . . 6 ⊢ ([(𝑓 ↾ (𝑀...𝑛)) / 𝑔]𝜓 ↔ 𝜒)
242		reseq1 5311	. . . . . . . 8 ⊢ (𝑓 = (𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) → (𝑓 ↾ (𝑀...𝑛)) = ((𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑛)))
243		fzssuz 12253	. . . . . . . . . 10 ⊢ (𝑀...𝑛) ⊆ (ℤ_≥‘𝑀)
244	243, 64	sseqtr4i 3601	. . . . . . . . 9 ⊢ (𝑀...𝑛) ⊆ 𝑍
245		resmpt 5369	. . . . . . . . 9 ⊢ ((𝑀...𝑛) ⊆ 𝑍 → ((𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑛)) = (𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)))
246	244, 245	ax-mp 5	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) ↾ (𝑀...𝑛)) = (𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚))
247	242, 246	syl6eq 2660	. . . . . . 7 ⊢ (𝑓 = (𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) → (𝑓 ↾ (𝑀...𝑛)) = (𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)))
248	247	sbceq1d 3407	. . . . . 6 ⊢ (𝑓 = (𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) → ([(𝑓 ↾ (𝑀...𝑛)) / 𝑔]𝜓 ↔ [(𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓))
249	241, 248	syl5bbr 273	. . . . 5 ⊢ (𝑓 = (𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) → (𝜒 ↔ [(𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓))
250	249	ralbidv 2969	. . . 4 ⊢ (𝑓 = (𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) → (∀𝑛 ∈ 𝑍 𝜒 ↔ ∀𝑛 ∈ 𝑍 [(𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓))
251	237, 250	anbi12d 743	. . 3 ⊢ (𝑓 = (𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)) → ((𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒) ↔ ((𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)):𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 [(𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓)))
252	236, 251	spcev 3273	. 2 ⊢ (((𝑚 ∈ 𝑍 ↦ ((𝐺‘𝑚)‘𝑚)):𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 [(𝑚 ∈ (𝑀...𝑛) ↦ ((𝐺‘𝑚)‘𝑚)) / 𝑔]𝜓) → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒))
253	224, 235, 252	syl2anc 691	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 {cab 2596 ∀wral 2896 ∃wrex 2897 Vcvv 3173 [wsbc 3402 ⊆ wss 3540 {csn 4125 ↦ cmpt 4643 dom cdm 5038 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ↑_𝑚 cmap 7744 1c1 9816 + caddc 9818 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198

This theorem is referenced by: sdclem1 32709

W3C validator