Theorem tfrlem12 8411

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 8406	. . . . 5 ⊢ Ord dom recs(𝐹)
3	2	a1i 11	. . . 4 ⊢ (recs(𝐹) ∈ V → Ord dom recs(𝐹))
4		dmexg 7905	. . . 4 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
5		elon2 6374	. . . 4 ⊢ (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
6	3, 4, 5	sylanbrc 583	. . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
7		onsuc 7813	. . . 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 8409	. . . 4 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
10	1, 8	tfrlem11 8410	. . . . . 6 ⊢ (dom recs(𝐹) ∈ On → (𝑧 ∈ suc dom recs(𝐹) → (𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧))))
11	10	ralrimiv 3132	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → ∀𝑧 ∈ suc dom recs(𝐹)(𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧)))
12		fveq2 6886	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐶‘𝑧) = (𝐶‘𝑦))
13		reseq2 5972	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐶 ↾ 𝑧) = (𝐶 ↾ 𝑦))
14	13	fveq2d 6890	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐹‘(𝐶 ↾ 𝑧)) = (𝐹‘(𝐶 ↾ 𝑦)))
15	12, 14	eqeq12d 2750	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧)) ↔ (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
16	15	cbvralvw 3223	. . . . 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 218	. . . 4 ⊢ (dom recs(𝐹) ∈ On → ∀𝑦 ∈ suc dom recs(𝐹)(𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))
18		fneq2 6640	. . . . . 6 ⊢ (𝑥 = suc dom recs(𝐹) → (𝐶 Fn 𝑥 ↔ 𝐶 Fn suc dom recs(𝐹)))
19		raleq 3306	. . . . . 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 632	. . . . 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 3605	. . . 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 836	. . 3 ⊢ (dom recs(𝐹) ∈ On → ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
23	6, 22	syl 17	. 2 ⊢ (recs(𝐹) ∈ V → ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
24		snex 5416	. . . . 5 ⊢ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V
25		unexg 7745	. . . . 5 ⊢ ((recs(𝐹) ∈ V ∧ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V) → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
26	24, 25	mpan2 691	. . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
27	8, 26	eqeltrid 2837	. . 3 ⊢ (recs(𝐹) ∈ V → 𝐶 ∈ V)
28		fneq1 6639	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓 Fn 𝑥 ↔ 𝐶 Fn 𝑥))
29		fveq1 6885	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝑓‘𝑦) = (𝐶‘𝑦))
30		reseq1 5971	. . . . . . . . 9 ⊢ (𝑓 = 𝐶 → (𝑓 ↾ 𝑦) = (𝐶 ↾ 𝑦))
31	30	fveq2d 6890	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝐶 ↾ 𝑦)))
32	29, 31	eqeq12d 2750	. . . . . . 7 ⊢ (𝑓 = 𝐶 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
33	32	ralbidv 3165	. . . . . 6 ⊢ (𝑓 = 𝐶 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
34	28, 33	anbi12d 632	. . . . 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 3166	. . . 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 3663	. . 3 ⊢ (𝐶 ∈ V → (𝐶 ∈ 𝐴 ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))))
37	27, 36	syl 17	. 2 ⊢ (recs(𝐹) ∈ V → (𝐶 ∈ 𝐴 ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))))
38	23, 37	mpbird 257	1 ⊢ (recs(𝐹) ∈ V → 𝐶 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2712  ∀wral 3050  ∃wrex 3059  Vcvv 3463   ∪ cun 3929  {csn 4606  ⟨cop 4612  dom cdm 5665   ↾ cres 5667  Ord word 6362  Oncon0 6363  suc csuc 6365   Fn wfn 6536  ‘cfv 6541  recscrecs 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fo 6547  df-fv 6549  df-ov 7416  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393

This theorem is referenced by:  tfrlem13  8412