Theorem fin23lem35 10344

Description: Lemma for fin23 10386. 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 10343	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))
7		fvex 6903	. . . . . . 7 ⊢ (𝑌‘𝐴) ∈ V
8		f1eq1 6781	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (𝑗:ω–1-1→V ↔ (𝑌‘𝐴):ω–1-1→V))
9		rneq 5934	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑌‘𝐴) → ran 𝑗 = ran (𝑌‘𝐴))
10	9	unieqd 4921	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → ∪ ran 𝑗 = ∪ ran (𝑌‘𝐴))
11	10	sseq1d 4012	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (∪ ran 𝑗 ⊆ 𝐺 ↔ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))
12	8, 11	anbi12d 629	. . . . . . . 8 ⊢ (𝑗 = (𝑌‘𝐴) → ((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) ↔ ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺)))
13		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → (𝑖‘𝑗) = (𝑖‘(𝑌‘𝐴)))
14		f1eq1 6781	. . . . . . . . . 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 5936	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑌‘𝐴) → ran (𝑖‘𝑗) = ran (𝑖‘(𝑌‘𝐴)))
17	16	unieqd 4921	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → ∪ ran (𝑖‘𝑗) = ∪ ran (𝑖‘(𝑌‘𝐴)))
18	17, 10	psseq12d 4093	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗 ↔ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
19	15, 18	anbi12d 629	. . . . . . . 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 343	. . . . . . 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 3594	. . . . . 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 479	. . . 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 494	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))
26		frsuc 8439	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴)))
27	26	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴)))
28	5	fveq1i 6891	. . . . . 6 ⊢ (𝑌‘suc 𝐴) = ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴)
29	5	fveq1i 6891	. . . . . . 7 ⊢ (𝑌‘𝐴) = ((rec(𝑖, ℎ) ↾ ω)‘𝐴)
30	29	fveq2i 6893	. . . . . 6 ⊢ (𝑖‘(𝑌‘𝐴)) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴))
31	27, 28, 30	3eqtr4g 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝑌‘suc 𝐴) = (𝑖‘(𝑌‘𝐴)))
32	31	rneqd 5936	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌‘𝐴)))
33	32	unieqd 4921	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) = ∪ ran (𝑖‘(𝑌‘𝐴)))
34	33	psseq1d 4091	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (∪ ran (𝑌‘suc 𝐴) ⊊ ∪ ran (𝑌‘𝐴) ↔ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
35	25, 34	mpbird 256	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) ⊊ ∪ ran (𝑌‘𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1537   = wceq 1539   ∈ wcel 2104  {cab 2707  ∀wral 3059  Vcvv 3472   ⊆ wss 3947   ⊊ wpss 3948  𝒫 cpw 4601  ∪ cuni 4907  ∩ cint 4949  ran crn 5676   ↾ cres 5677  suc csuc 6365  –1-1→wf1 6539  ‘cfv 6542  (class class class)co 7411  ωcom 7857  reccrdg 8411   ↑_m cmap 8822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412

This theorem is referenced by:  fin23lem36  10345  fin23lem38  10346  fin23lem39  10347