Theorem tfrlem12 8435

Description: Lemma for transfinite recursion. Show 𝐶 is an acceptable function. (Contributed by NM, 15-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

Hypotheses

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
tfrlem.3	⊢ 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})

Assertion

Ref	Expression
tfrlem12	⊢ (recs(𝐹) ∈ V → 𝐶 ∈ 𝐴)

Distinct variable groups:   𝑥,𝑓,𝑦,𝐶   𝑓,𝐹,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑓)

Proof of Theorem tfrlem12

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . . . 6 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem8 8430	. . . . 5 ⊢ Ord dom recs(𝐹)
3	2	a1i 11	. . . 4 ⊢ (recs(𝐹) ∈ V → Ord dom recs(𝐹))
4		dmexg 7947	. . . 4 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
5		elon2 6399	. . . 4 ⊢ (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
6	3, 4, 5	sylanbrc 595	. . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
7		onsuc 7858	. . . 4 ⊢ (dom recs(𝐹) ∈ On → suc dom recs(𝐹) ∈ On)
8		tfrlem.3	. . . . 5 ⊢ 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
9	1, 8	tfrlem10 8433	. . . 4 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
10	1, 8	tfrlem11 8434	. . . . . 6 ⊢ (dom recs(𝐹) ∈ On → (𝑧 ∈ suc dom recs(𝐹) → (𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧))))
11	10	ralrimiv 3157	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → ∀𝑧 ∈ suc dom recs(𝐹)(𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧)))
12		fveq2 6916	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐶‘𝑧) = (𝐶‘𝑦))
13		reseq2 5981	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐶 ↾ 𝑧) = (𝐶 ↾ 𝑦))
14	13	fveq2d 6920	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐹‘(𝐶 ↾ 𝑧)) = (𝐹‘(𝐶 ↾ 𝑦)))
15	12, 14	eqeq12d 2779	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧)) ↔ (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
16	15	cbvralvw 3244	. . . . 5 ⊢ (∀𝑧 ∈ suc dom recs(𝐹)(𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧)) ↔ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))
17	11, 16	sylib 221	. . . 4 ⊢ (dom recs(𝐹) ∈ On → ∀𝑦 ∈ suc dom recs(𝐹)(𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))
18		fneq2 6661	. . . . . 6 ⊢ (𝑥 = suc dom recs(𝐹) → (𝐶 Fn 𝑥 ↔ 𝐶 Fn suc dom recs(𝐹)))
19		raleq 3320	. . . . . 6 ⊢ (𝑥 = suc dom recs(𝐹) → (∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)) ↔ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
20	18, 19	anbi12d 644	. . . . 5 ⊢ (𝑥 = suc dom recs(𝐹) → ((𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))) ↔ (𝐶 Fn suc dom recs(𝐹) ∧ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))))
21	20	rspcev 3580	. . . 4 ⊢ ((suc dom recs(𝐹) ∈ On ∧ (𝐶 Fn suc dom recs(𝐹) ∧ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))) → ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
22	7, 9, 17, 21	syl12anc 850	. . 3 ⊢ (dom recs(𝐹) ∈ On → ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
23	6, 22	syl 18	. 2 ⊢ (recs(𝐹) ∈ V → ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
24		snex 5404	. . . . 5 ⊢ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V
25		unexg 7792	. . . . 5 ⊢ ((recs(𝐹) ∈ V ∧ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V) → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
26	24, 25	mpan2 704	. . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
27	8, 26	eqeltrid 2868	. . 3 ⊢ (recs(𝐹) ∈ V → 𝐶 ∈ V)
28		fneq1 6660	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓 Fn 𝑥 ↔ 𝐶 Fn 𝑥))
29		fveq1 6915	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝑓‘𝑦) = (𝐶‘𝑦))
30		reseq1 5980	. . . . . . . . 9 ⊢ (𝑓 = 𝐶 → (𝑓 ↾ 𝑦) = (𝐶 ↾ 𝑦))
31	30	fveq2d 6920	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝐶 ↾ 𝑦)))
32	29, 31	eqeq12d 2779	. . . . . . 7 ⊢ (𝑓 = 𝐶 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
33	32	ralbidv 3189	. . . . . 6 ⊢ (𝑓 = 𝐶 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
34	28, 33	anbi12d 644	. . . . 5 ⊢ (𝑓 = 𝐶 → ((𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) ↔ (𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))))
35	34	rexbidv 3190	. . . 4 ⊢ (𝑓 = 𝐶 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))))
36	35, 1	elab2g 3637	. . 3 ⊢ (𝐶 ∈ V → (𝐶 ∈ 𝐴 ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))))
37	27, 36	syl 18	. 2 ⊢ (recs(𝐹) ∈ V → (𝐶 ∈ 𝐴 ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))))
38	23, 37	mpbird 260	1 ⊢ (recs(𝐹) ∈ V → 𝐶 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ∪ cun 4152  {csn 4588  ⟨cop 4840  dom cdm 5658   ↾ cres 5660  Ord word 6387  Oncon0 6388  suc csuc 6390   Fn wfn 6561  ‘cfv 6566  recscrecs 8416

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 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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-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 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-ov 7454  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417

This theorem is used by:  tfrlem13  8436