Theorem fin23lem35 10449

Description: Lemma for fin23 10491. 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⟶1-1V)
fin23lem.g	⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
fin23lem.h	⊢ (𝜑 → ∀𝑗((𝑗:ω⟶1-1V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω⟶1-1V ∧ ∪ 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⟶1-1V)
3		fin23lem.g	. . . . 5 ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
4		fin23lem.h	. . . . 5 ⊢ (𝜑 → ∀𝑗((𝑗:ω⟶1-1V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω⟶1-1V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
5		fin23lem.i	. . . . 5 ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)
6	1, 2, 3, 4, 5	fin23lem34 10448	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑌‘𝐴):ω⟶1-1V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))
7		fvex 6918	. . . . . . 7 ⊢ (𝑌‘𝐴) ∈ V
8		f1eq1 6792	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (𝑗:ω⟶1-1V ↔ (𝑌‘𝐴):ω⟶1-1V))
9		rneq 5924	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑌‘𝐴) → ran 𝑗 = ran (𝑌‘𝐴))
10	9	unieqd 4883	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → ∪ ran 𝑗 = ∪ ran (𝑌‘𝐴))
11	10	sseq1d 3944	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → (∪ ran 𝑗 ⊆ 𝐺 ↔ ∪ ran (𝑌‘𝐴) ⊆ 𝐺))
12	8, 11	anbi12d 644	. . . . . . . 8 ⊢ (𝑗 = (𝑌‘𝐴) → ((𝑗:ω⟶1-1V ∧ ∪ ran 𝑗 ⊆ 𝐺) ↔ ((𝑌‘𝐴):ω⟶1-1V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺)))
13		fveq2 6905	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → (𝑖‘𝑗) = (𝑖‘(𝑌‘𝐴)))
14		f1eq1 6792	. . . . . . . . . 10 ⊢ ((𝑖‘𝑗) = (𝑖‘(𝑌‘𝐴)) → ((𝑖‘𝑗):ω⟶1-1V ↔ (𝑖‘(𝑌‘𝐴)):ω⟶1-1V))
15	13, 14	syl 18	. . . . . . . . 9 ⊢ (𝑗 = (𝑌‘𝐴) → ((𝑖‘𝑗):ω⟶1-1V ↔ (𝑖‘(𝑌‘𝐴)):ω⟶1-1V))
16	13	rneqd 5926	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑌‘𝐴) → ran (𝑖‘𝑗) = ran (𝑖‘(𝑌‘𝐴)))
17	16	unieqd 4883	. . . . . . . . . 10 ⊢ (𝑗 = (𝑌‘𝐴) → ∪ ran (𝑖‘𝑗) = ∪ ran (𝑖‘(𝑌‘𝐴)))
18	17, 10	psseq12d 4027	. . . . . . . . 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⟶1-1V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗) ↔ ((𝑖‘(𝑌‘𝐴)):ω⟶1-1V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
20	12, 19	imbi12d 347	. . . . . . 7 ⊢ (𝑗 = (𝑌‘𝐴) → (((𝑗:ω⟶1-1V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω⟶1-1V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)) ↔ (((𝑌‘𝐴):ω⟶1-1V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω⟶1-1V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))))
21	7, 20	spcv 3563	. . . . . 6 ⊢ (∀𝑗((𝑗:ω⟶1-1V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω⟶1-1V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)) → (((𝑌‘𝐴):ω⟶1-1V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω⟶1-1V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
22	4, 21	syl 18	. . . . 5 ⊢ (𝜑 → (((𝑌‘𝐴):ω⟶1-1V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω⟶1-1V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
23	22	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (((𝑌‘𝐴):ω⟶1-1V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌‘𝐴)):ω⟶1-1V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))))
24	6, 23	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑖‘(𝑌‘𝐴)):ω⟶1-1V ∧ ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴)))
25	24	simprd 501	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑖‘(𝑌‘𝐴)) ⊊ ∪ ran (𝑌‘𝐴))
26		frsuc 8470	. . . . . . 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 6906	. . . . . 6 ⊢ (𝑌‘suc 𝐴) = ((rec(𝑖, ℎ) ↾ ω)‘suc 𝐴)
29	5	fveq1i 6906	. . . . . . 7 ⊢ (𝑌‘𝐴) = ((rec(𝑖, ℎ) ↾ ω)‘𝐴)
30	29	fveq2i 6908	. . . . . 6 ⊢ (𝑖‘(𝑌‘𝐴)) = (𝑖‘((rec(𝑖, ℎ) ↾ ω)‘𝐴))
31	27, 28, 30	3eqtr4g 2823	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝑌‘suc 𝐴) = (𝑖‘(𝑌‘𝐴)))
32	31	rneqd 5926	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌‘𝐴)))
33	32	unieqd 4883	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) = ∪ ran (𝑖‘(𝑌‘𝐴)))
34	33	psseq1d 4025	. 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 1568   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ⊊ wpss 4016  𝒫 cpw 4560  ∪ cuni 4870  ∩ cint 4910  ran crn 5656   ↾ cres 5657  suc csuc 6379  ⟶1-1wf1 6552  ‘cfv 6555  (class class class)co 7440  ωcom 7898  reccrdg 8442   ↑_m cmap 8872

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443

This theorem is used by:  fin23lem36  10450  fin23lem38  10451  fin23lem39  10452