Theorem fin23lem35 9052

Description: Lemma for fin23 9094. Strict order property of 𝑌. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Hypotheses

Ref	Expression
fin23lem33.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
fin23lem.f	⊢ (𝜑 → ℎ:ω–1-1→V)
fin23lem.g	⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
fin23lem.h	⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
fin23lem.i	⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)

Assertion

Ref	Expression
fin23lem35	⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) ⊊ ∪ ran (𝑌‘𝐴))

Distinct variable groups:   𝑔,𝑎,𝑖,𝑗,𝑥   𝐴,𝑎,𝑗   ℎ,𝑎,𝐺,𝑔,𝑖,𝑗,𝑥   𝐹,𝑎   𝜑,𝑎,𝑗   𝑌,𝑎,𝑗

Allowed substitution hints:   𝜑(𝑥,𝑔,ℎ,𝑖)   𝐴(𝑥,𝑔,ℎ,𝑖)   𝐹(𝑥,𝑔,ℎ,𝑖,𝑗)   𝑌(𝑥,𝑔,ℎ,𝑖)

Proof of Theorem fin23lem35

Step	Hyp	Ref	Expression
1		fin23lem33.f	. . . . 5 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
2		fin23lem.f	. . . . 5 ⊢ (𝜑 → ℎ:ω–1-1→V)
3		fin23lem.g	. . . . 5 ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
4		fin23lem.h	. . . . 5 ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
5		fin23lem.i	. . . . 5 ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)
6	1, 2, 3, 4, 5	fin23lem34 9051	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))
7		fvex 6113	. . . . . . 7 ⊢ (𝑌‘𝐴) ∈ V
8		f1eq1 6009	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (𝑗:ω–1-1→V ↔ (𝑌‘𝐴):ω–1-1→V))
9		rneq 5272	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑌‘𝐴) → ran 𝑗 = ran (𝑌‘𝐴))
10	9	unieqd 4382	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → ∪ ran 𝑗 = ∪ ran (𝑌‘𝐴))
11	10	sseq1d 3595	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (∪ ran 𝑗 ⊆ 𝐺 ↔ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))
12	8, 11	anbi12d 743	. . . . . . . 8 ⊢ (𝑗 = (𝑌‘𝐴) → ((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) ↔ ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺)))
13		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → (𝑖‘𝑗) = (𝑖‘(𝑌‘𝐴)))
14		f1eq1 6009	. . . . . . . . . 10 ⊢ ((𝑖‘𝑗) = (𝑖‘(𝑌‘𝐴)) → ((𝑖‘𝑗):ω–1-1→V ↔ (𝑖‘(𝑌‘𝐴)):ω–1-1→V))
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → ((𝑖‘𝑗):ω–1-1→V ↔ (𝑖‘(𝑌‘𝐴)):ω–1-1→V))
16	13	rneqd 5274	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑌‘𝐴) → ran (𝑖‘𝑗) = ran (𝑖‘(𝑌‘𝐴)))
17	16	unieqd 4382	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → ∪ ran (𝑖‘𝑗) = ∪ ran (𝑖‘(𝑌‘𝐴)))
18	17, 10	psseq12d 3663	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗 ↔ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
19	15, 18	anbi12d 743	. . . . . . . 8 ⊢ (𝑗 = (𝑌‘𝐴) → (((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗) ↔ ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
20	12, 19	imbi12d 333	. . . . . . 7 ⊢ (𝑗 = (𝑌‘𝐴) → (((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)) ↔ (((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))))
21	7, 20	spcv 3272	. . . . . 6 ⊢ (∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)) → (((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
22	4, 21	syl 17	. . . . 5 ⊢ (𝜑 → (((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
23	22	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
24	6, 23	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
25	24	simprd 478	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))
26		frsuc 7419	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴)))
27	26	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴)))
28	5	fveq1i 6104	. . . . . 6 ⊢ (𝑌‘suc 𝐴) = ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴)
29	5	fveq1i 6104	. . . . . . 7 ⊢ (𝑌‘𝐴) = ((rec(𝑖, ℎ) ↾ ω)‘𝐴)
30	29	fveq2i 6106	. . . . . 6 ⊢ (𝑖‘(𝑌‘𝐴)) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴))
31	27, 28, 30	3eqtr4g 2669	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝑌‘suc 𝐴) = (𝑖‘(𝑌‘𝐴)))
32	31	rneqd 5274	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌‘𝐴)))
33	32	unieqd 4382	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) = ∪ ran (𝑖‘(𝑌‘𝐴)))
34	33	psseq1d 3661	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (∪ ran (𝑌‘suc 𝐴) ⊊ ∪ ran (𝑌‘𝐴) ↔ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
35	25, 34	mpbird 246	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) ⊊ ∪ ran (𝑌‘𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ⊆ wss 3540   ⊊ wpss 3541  𝒫 cpw 4108  ∪ cuni 4372  ∩ cint 4410  ran crn 5039   ↾ cres 5040  suc csuc 5642  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  ωcom 6957  reccrdg 7392   ↑_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-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-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393

This theorem is referenced by:  fin23lem36  9053  fin23lem38  9054  fin23lem39  9055