Theorem tfrlem12 7638

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 7633	. . . . 5 ⊢ Ord dom recs(𝐹)
3	2	a1i 11	. . . 4 ⊢ (recs(𝐹) ∈ V → Ord dom recs(𝐹))
4		dmexg 7244	. . . 4 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
5		elon2 5877	. . . 4 ⊢ (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
6	3, 4, 5	sylanbrc 572	. . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
7		suceloni 7160	. . . 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 7636	. . . 4 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
10	1, 8	tfrlem11 7637	. . . . . 6 ⊢ (dom recs(𝐹) ∈ On → (𝑧 ∈ suc dom recs(𝐹) → (𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧))))
11	10	ralrimiv 3114	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → ∀𝑧 ∈ suc dom recs(𝐹)(𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧)))
12		fveq2 6332	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐶‘𝑧) = (𝐶‘𝑦))
13		reseq2 5529	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐶 ↾ 𝑧) = (𝐶 ↾ 𝑦))
14	13	fveq2d 6336	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐹‘(𝐶 ↾ 𝑧)) = (𝐹‘(𝐶 ↾ 𝑦)))
15	12, 14	eqeq12d 2786	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧)) ↔ (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
16	15	cbvralv 3320	. . . . 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 208	. . . 4 ⊢ (dom recs(𝐹) ∈ On → ∀𝑦 ∈ suc dom recs(𝐹)(𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))
18		fneq2 6120	. . . . . 6 ⊢ (𝑥 = suc dom recs(𝐹) → (𝐶 Fn 𝑥 ↔ 𝐶 Fn suc dom recs(𝐹)))
19		raleq 3287	. . . . . 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 616	. . . . 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 3460	. . . 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 1474	. . 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 5036	. . . . 5 ⊢ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V
25		unexg 7106	. . . . 5 ⊢ ((recs(𝐹) ∈ V ∧ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V) → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
26	24, 25	mpan2 671	. . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
27	8, 26	syl5eqel 2854	. . 3 ⊢ (recs(𝐹) ∈ V → 𝐶 ∈ V)
28		fneq1 6119	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓 Fn 𝑥 ↔ 𝐶 Fn 𝑥))
29		fveq1 6331	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝑓‘𝑦) = (𝐶‘𝑦))
30		reseq1 5528	. . . . . . . . 9 ⊢ (𝑓 = 𝐶 → (𝑓 ↾ 𝑦) = (𝐶 ↾ 𝑦))
31	30	fveq2d 6336	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝐶 ↾ 𝑦)))
32	29, 31	eqeq12d 2786	. . . . . . 7 ⊢ (𝑓 = 𝐶 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
33	32	ralbidv 3135	. . . . . 6 ⊢ (𝑓 = 𝐶 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
34	28, 33	anbi12d 616	. . . . 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 3200	. . . 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 3504	. . 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 247	1 ⊢ (recs(𝐹) ∈ V → 𝐶 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {cab 2757  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ∪ cun 3721  {csn 4316  ⟨cop 4322  dom cdm 5249   ↾ cres 5251  Ord word 5865  Oncon0 5866  suc csuc 5868   Fn wfn 6026  ‘cfv 6031  recscrecs 7620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-wrecs 7559  df-recs 7621

This theorem is referenced by:  tfrlem13  7639