Theorem tfrlem12 8191

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 8186	. . . . 5 ⊢ Ord dom recs(𝐹)
3	2	a1i 11	. . . 4 ⊢ (recs(𝐹) ∈ V → Ord dom recs(𝐹))
4		dmexg 7724	. . . 4 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
5		elon2 6262	. . . 4 ⊢ (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
6	3, 4, 5	sylanbrc 582	. . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
7		suceloni 7635	. . . 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 8189	. . . 4 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
10	1, 8	tfrlem11 8190	. . . . . 6 ⊢ (dom recs(𝐹) ∈ On → (𝑧 ∈ suc dom recs(𝐹) → (𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧))))
11	10	ralrimiv 3106	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → ∀𝑧 ∈ suc dom recs(𝐹)(𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧)))
12		fveq2 6756	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐶‘𝑧) = (𝐶‘𝑦))
13		reseq2 5875	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐶 ↾ 𝑧) = (𝐶 ↾ 𝑦))
14	13	fveq2d 6760	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝐹‘(𝐶 ↾ 𝑧)) = (𝐹‘(𝐶 ↾ 𝑦)))
15	12, 14	eqeq12d 2754	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝐶‘𝑧) = (𝐹‘(𝐶 ↾ 𝑧)) ↔ (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
16	15	cbvralvw 3372	. . . . 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 6509	. . . . . 6 ⊢ (𝑥 = suc dom recs(𝐹) → (𝐶 Fn 𝑥 ↔ 𝐶 Fn suc dom recs(𝐹)))
19		raleq 3333	. . . . . 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 3552	. . . 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 833	. . 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 5349	. . . . 5 ⊢ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V
25		unexg 7577	. . . . 5 ⊢ ((recs(𝐹) ∈ V ∧ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V) → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
26	24, 25	mpan2 687	. . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
27	8, 26	eqeltrid 2843	. . 3 ⊢ (recs(𝐹) ∈ V → 𝐶 ∈ V)
28		fneq1 6508	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓 Fn 𝑥 ↔ 𝐶 Fn 𝑥))
29		fveq1 6755	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝑓‘𝑦) = (𝐶‘𝑦))
30		reseq1 5874	. . . . . . . . 9 ⊢ (𝑓 = 𝐶 → (𝑓 ↾ 𝑦) = (𝐶 ↾ 𝑦))
31	30	fveq2d 6760	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝐶 ↾ 𝑦)))
32	29, 31	eqeq12d 2754	. . . . . . 7 ⊢ (𝑓 = 𝐶 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝐶‘𝑦) = (𝐹‘(𝐶 ↾ 𝑦))))
33	32	ralbidv 3120	. . . . . 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 3225	. . . 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 3604	. . 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 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∪ cun 3881  {csn 4558  ⟨cop 4564  dom cdm 5580   ↾ cres 5582  Ord word 6250  Oncon0 6251  suc csuc 6253   Fn wfn 6413  ‘cfv 6418  recscrecs 8172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-ov 7258  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173

This theorem is referenced by:  tfrlem13  8192