Theorem fseqenlem2 8731

Description: Lemma for fseqen 8733. (Contributed by Mario Carneiro, 17-May-2015.)

Hypotheses

Ref	Expression
fseqenlem.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fseqenlem.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)
fseqenlem.f	⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)
fseqenlem.g	⊢ 𝐺 = seq𝜔((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})
fseqenlem.k	⊢ 𝐾 = (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)

Assertion

Ref	Expression
fseqenlem2	⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)–1-1→(ω × 𝐴))

Distinct variable groups:   𝑦,𝐵   𝑓,𝑛,𝑥,𝐹   𝑦,𝑘,𝐺   𝑓,𝑘,𝑦,𝐴,𝑛,𝑥   𝜑,𝑘,𝑛,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑥,𝑓,𝑘,𝑛)   𝐹(𝑦,𝑘)   𝐺(𝑥,𝑓,𝑛)   𝐾(𝑥,𝑦,𝑓,𝑘,𝑛)   𝑉(𝑥,𝑦,𝑓,𝑘,𝑛)

Proof of Theorem fseqenlem2

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4460	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴 ↑𝑚 𝑘))
2		elmapi 7765	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ↑𝑚 𝑘) → 𝑦:𝑘⟶𝐴)
3	2	ad2antll 761	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → 𝑦:𝑘⟶𝐴)
4		fdm 5964	. . . . . . . . 9 ⊢ (𝑦:𝑘⟶𝐴 → dom 𝑦 = 𝑘)
5	3, 4	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → dom 𝑦 = 𝑘)
6		simprl 790	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → 𝑘 ∈ ω)
7	5, 6	eqeltrd 2688	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → dom 𝑦 ∈ ω)
8	5	fveq2d 6107	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → (𝐺‘dom 𝑦) = (𝐺‘𝑘))
9	8	fveq1d 6105	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘𝑘)‘𝑦))
10		fseqenlem.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
11		fseqenlem.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
12		fseqenlem.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)
13		fseqenlem.g	. . . . . . . . . . . 12 ⊢ 𝐺 = seq𝜔((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})
14	10, 11, 12, 13	fseqenlem1 8730	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → (𝐺‘𝑘):(𝐴 ↑𝑚 𝑘)–1-1→𝐴)
15	14	adantrr 749	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → (𝐺‘𝑘):(𝐴 ↑𝑚 𝑘)–1-1→𝐴)
16		f1f 6014	. . . . . . . . . 10 ⊢ ((𝐺‘𝑘):(𝐴 ↑𝑚 𝑘)–1-1→𝐴 → (𝐺‘𝑘):(𝐴 ↑𝑚 𝑘)⟶𝐴)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → (𝐺‘𝑘):(𝐴 ↑𝑚 𝑘)⟶𝐴)
18		simprr 792	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → 𝑦 ∈ (𝐴 ↑𝑚 𝑘))
19	17, 18	ffvelrnd 6268	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → ((𝐺‘𝑘)‘𝑦) ∈ 𝐴)
20	9, 19	eqeltrd 2688	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
21		opelxpi 5072	. . . . . . 7 ⊢ ((dom 𝑦 ∈ ω ∧ ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
22	7, 20, 21	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
23	22	rexlimdvaa 3014	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴 ↑𝑚 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
24	1, 23	syl5bi 231	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
25	24	imp 444	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
26		fseqenlem.k	. . 3 ⊢ 𝐾 = (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)
27	25, 26	fmptd 6292	. 2 ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)⟶(ω × 𝐴))
28		ffun 5961	. . . . . . . . . . . . . . 15 ⊢ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
29		funbrfv2b 6150	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐾 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾‘𝑧) = 𝑤)))
30	27, 28, 29	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾‘𝑧) = 𝑤)))
31	30	simplbda 652	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐾‘𝑧) = 𝑤)
32	30	simprbda 651	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ dom 𝐾)
33		fdm 5964	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)⟶(ω × 𝐴) → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘))
34	27, 33	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘))
35	34	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘))
36	32, 35	eleqtrd 2690	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘))
37		dmeq 5246	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
38	37	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
39		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
40	38, 39	fveq12d 6109	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
41	37, 40	opeq12d 4348	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
42		opex 4859	. . . . . . . . . . . . . . 15 ⊢ ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
43	41, 26, 42	fvmpt 6191	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) → (𝐾‘𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
44	36, 43	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐾‘𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
45	31, 44	eqtr3d 2646	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
46	45	fveq2d 6107	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1st ‘𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
47		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
48	47	dmex 6991	. . . . . . . . . . . 12 ⊢ dom 𝑧 ∈ V
49		fvex 6113	. . . . . . . . . . . 12 ⊢ ((𝐺‘dom 𝑧)‘𝑧) ∈ V
50	48, 49	op1st 7067	. . . . . . . . . . 11 ⊢ (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
51	46, 50	syl6eq 2660	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1st ‘𝑤) = dom 𝑧)
52	51	fveq2d 6107	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘(1st ‘𝑤)) = (𝐺‘dom 𝑧))
53	52	cnveqd 5220	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → ◡(𝐺‘(1st ‘𝑤)) = ◡(𝐺‘dom 𝑧))
54	45	fveq2d 6107	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2nd ‘𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
55	48, 49	op2nd 7068	. . . . . . . . 9 ⊢ (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
56	54, 55	syl6eq 2660	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2nd ‘𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
57	53, 56	fveq12d 6109	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤)) = (◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
58		eliun 4460	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴 ↑𝑚 𝑘))
59		elmapi 7765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝐴 ↑𝑚 𝑘) → 𝑧:𝑘⟶𝐴)
60	59	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → 𝑧:𝑘⟶𝐴)
61		fdm 5964	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧:𝑘⟶𝐴 → dom 𝑧 = 𝑘)
62	60, 61	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → dom 𝑧 = 𝑘)
63		simpl 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → 𝑘 ∈ ω)
64	62, 63	eqeltrd 2688	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → dom 𝑧 ∈ ω)
65		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → 𝑧 ∈ (𝐴 ↑𝑚 𝑘))
66	62	oveq2d 6565	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → (𝐴 ↑𝑚 dom 𝑧) = (𝐴 ↑𝑚 𝑘))
67	65, 66	eleqtrrd 2691	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧))
68	64, 67	jca 553	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧)))
69	68	rexlimiva 3010	. . . . . . . . . . . . 13 ⊢ (∃𝑘 ∈ ω 𝑧 ∈ (𝐴 ↑𝑚 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧)))
70	58, 69	sylbi 206	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧)))
71	36, 70	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧)))
72	71	simpld 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → dom 𝑧 ∈ ω)
73	10, 11, 12, 13	fseqenlem1 8730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴 ↑𝑚 dom 𝑧)–1-1→𝐴)
74	72, 73	syldan 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴 ↑𝑚 dom 𝑧)–1-1→𝐴)
75		f1f1orn 6061	. . . . . . . . 9 ⊢ ((𝐺‘dom 𝑧):(𝐴 ↑𝑚 dom 𝑧)–1-1→𝐴 → (𝐺‘dom 𝑧):(𝐴 ↑𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
76	74, 75	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴 ↑𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
77	71	simprd 478	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧))
78		f1ocnvfv1 6432	. . . . . . . 8 ⊢ (((𝐺‘dom 𝑧):(𝐴 ↑𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧) ∧ 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧)) → (◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
79	76, 77, 78	syl2anc 691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
80	57, 79	eqtr2d 2645	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤)))
81	80	ex 449	. . . . 5 ⊢ (𝜑 → (𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))))
82	81	alrimiv 1842	. . . 4 ⊢ (𝜑 → ∀𝑧(𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))))
83		mo2icl 3352	. . . 4 ⊢ (∀𝑧(𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
84	82, 83	syl 17	. . 3 ⊢ (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
85	84	alrimiv 1842	. 2 ⊢ (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
86		dff12 6013	. 2 ⊢ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)–1-1→(ω × 𝐴) ↔ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)⟶(ω × 𝐴) ∧ ∀𝑤∃*𝑧 𝑧𝐾𝑤))
87	27, 85, 86	sylanbrc 695	1 ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)–1-1→(ω × 𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459  ∃wrex 2897  Vcvv 3173  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040  suc csuc 5642  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  1^st c1st 7057  2^nd c2nd 7058  seq_𝜔cseqom 7429   ↑_𝑚 cmap 7744

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

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-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-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-seqom 7430  df-1o 7447  df-map 7746

This theorem is referenced by:  fseqen  8733  pwfseqlem5  9364