Theorem fin23lem35 10425

Description: Lemma for fin23 10467. 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 10424	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))
7		fvex 6898	. . . . . . 7 ⊢ (𝑌‘𝐴) ∈ V
8		f1eq1 6773	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (𝑗:ω–1-1→V ↔ (𝑌‘𝐴):ω–1-1→V))
9		rneq 5917	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑌‘𝐴) → ran 𝑗 = ran (𝑌‘𝐴))
10	9	unieqd 4879	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → ∪ ran 𝑗 = ∪ ran (𝑌‘𝐴))
11	10	sseq1d 3939	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (∪ ran 𝑗 ⊆ 𝐺 ↔ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))
12	8, 11	anbi12d 644	. . . . . . . 8 ⊢ (𝑗 = (𝑌‘𝐴) → ((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) ↔ ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺)))
13		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → (𝑖‘𝑗) = (𝑖‘(𝑌‘𝐴)))
14		f1eq1 6773	. . . . . . . . . 10 ⊢ ((𝑖‘𝑗) = (𝑖‘(𝑌‘𝐴)) → ((𝑖‘𝑗):ω–1-1→V ↔ (𝑖‘(𝑌‘𝐴)):ω–1-1→V))
15	13, 14	syl 18	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → ((𝑖‘𝑗):ω–1-1→V ↔ (𝑖‘(𝑌‘𝐴)):ω–1-1→V))
16	13	rneqd 5919	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑌‘𝐴) → ran (𝑖‘𝑗) = ran (𝑖‘(𝑌‘𝐴)))
17	16	unieqd 4879	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → ∪ ran (𝑖‘𝑗) = ∪ ran (𝑖‘(𝑌‘𝐴)))
18	17, 10	psseq12d 4024	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗 ↔ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
19	15, 18	anbi12d 644	. . . . . . . 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 347	. . . . . . 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 3559	. . . . . 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 18	. . . . 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 486	. . . 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 16	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑖‘(𝑌‘𝐴)):ω–1-1→V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
25	24	simprd 501	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))
26		frsuc 8445	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴)))
27	26	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴)))
28	5	fveq1i 6886	. . . . . 6 ⊢ (𝑌‘suc 𝐴) = ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴)
29	5	fveq1i 6886	. . . . . . 7 ⊢ (𝑌‘𝐴) = ((rec(𝑖, ℎ) ↾ ω)‘𝐴)
30	29	fveq2i 6888	. . . . . 6 ⊢ (𝑖‘(𝑌‘𝐴)) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴))
31	27, 28, 30	3eqtr4g 2820	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝑌‘suc 𝐴) = (𝑖‘(𝑌‘𝐴)))
32	31	rneqd 5919	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌‘𝐴)))
33	32	unieqd 4879	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) = ∪ ran (𝑖‘(𝑌‘𝐴)))
34	33	psseq1d 4022	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (∪ ran (𝑌‘suc 𝐴) ⊊ ∪ ran (𝑌‘𝐴) ↔ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
35	25, 34	mpbird 260	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) ⊊ ∪ ran (𝑌‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ⊊ wpss 4013  𝒫 cpw 4556  ∪ cuni 4866  ∩ cint 4906  ran crn 5651   ↾ cres 5652  suc csuc 6364  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7420  ωcom 7877  reccrdg 8417   ↑_m cmap 8847

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418

This theorem is used by:  fin23lem36  10426  fin23lem38  10427  fin23lem39  10428