Theorem tfrlem12 8419

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 8414	. . . . 5 ⊢ Ord dom recs(𝐹)
3	2	a1i 11	. . . 4 ⊢ (recs(𝐹) ∈ V → Ord dom recs(𝐹))
4		dmexg 7914	. . . 4 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
5		elon2 6387	. . . 4 ⊢ (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
6	3, 4, 5	sylanbrc 581	. . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
7		onsuc 7820	. . . 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 8417	. . . 4 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
10	1, 8	tfrlem11 8418	. . . . . 6 ⊢ (dom recs(𝐹) ∈ On → (𝑧 ∈ suc dom recs(𝐹) → (𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧))))
11	10	ralrimiv 3135	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → ∀𝑧 ∈ suc dom recs(𝐹)(𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧)))
12		fveq2 6901	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐶‘𝑧) = (𝐶‘𝑦))
13		reseq2 5984	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐶 ↾ 𝑧) = (𝐶 ↾ 𝑦))
14	13	fveq2d 6905	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐹‘(𝐶 ↾ 𝑧)) = (𝐹‘(𝐶 ↾ 𝑦)))
15	12, 14	eqeq12d 2742	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧)) ↔ (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
16	15	cbvralvw 3225	. . . . 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 217	. . . 4 ⊢ (dom recs(𝐹) ∈ On → ∀𝑦 ∈ suc dom recs(𝐹)(𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦)))
18		fneq2 6652	. . . . . 6 ⊢ (𝑥 = suc dom recs(𝐹) → (𝐶 Fn 𝑥 ↔ 𝐶 Fn suc dom recs(𝐹)))
19		raleq 3312	. . . . . 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 630	. . . . 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 3608	. . . 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 835	. . 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 5437	. . . . 5 ⊢ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V
25		unexg 7757	. . . . 5 ⊢ ((recs(𝐹) ∈ V ∧ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V) → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
26	24, 25	mpan2 689	. . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
27	8, 26	eqeltrid 2830	. . 3 ⊢ (recs(𝐹) ∈ V → 𝐶 ∈ V)
28		fneq1 6651	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓 Fn 𝑥 ↔ 𝐶 Fn 𝑥))
29		fveq1 6900	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝑓‘𝑦) = (𝐶‘𝑦))
30		reseq1 5983	. . . . . . . . 9 ⊢ (𝑓 = 𝐶 → (𝑓 ↾ 𝑦) = (𝐶 ↾ 𝑦))
31	30	fveq2d 6905	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝐶 ↾ 𝑦)))
32	29, 31	eqeq12d 2742	. . . . . . 7 ⊢ (𝑓 = 𝐶 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
33	32	ralbidv 3168	. . . . . 6 ⊢ (𝑓 = 𝐶 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
34	28, 33	anbi12d 630	. . . . 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 3169	. . . 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 3668	. . 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 256	1 ⊢ (recs(𝐹) ∈ V → 𝐶 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  {cab 2703  ∀wral 3051  ∃wrex 3060  Vcvv 3462   ∪ cun 3945  {csn 4633  ⟨cop 4639  dom cdm 5682   ↾ cres 5684  Ord word 6375  Oncon0 6376  suc csuc 6378   Fn wfn 6549  ‘cfv 6554  recscrecs 8400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fo 6560  df-fv 6562  df-ov 7427  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401

This theorem is referenced by:  tfrlem13  8420