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Theorem tfrlem12 8012
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 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlem.3 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
Assertion
Ref Expression
tfrlem12 (recs(𝐹) ∈ V → 𝐶𝐴)
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.
StepHypRef Expression
1 tfrlem.1 . . . . . 6 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem8 8007 . . . . 5 Ord dom recs(𝐹)
32a1i 11 . . . 4 (recs(𝐹) ∈ V → Ord dom recs(𝐹))
4 dmexg 7598 . . . 4 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
5 elon2 6174 . . . 4 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
63, 4, 5sylanbrc 586 . . 3 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
7 suceloni 7512 . . . 4 (dom recs(𝐹) ∈ On → suc dom recs(𝐹) ∈ On)
8 tfrlem.3 . . . . 5 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
91, 8tfrlem10 8010 . . . 4 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
101, 8tfrlem11 8011 . . . . . 6 (dom recs(𝐹) ∈ On → (𝑧 ∈ suc dom recs(𝐹) → (𝐶𝑧) = (𝐹‘(𝐶𝑧))))
1110ralrimiv 3151 . . . . 5 (dom recs(𝐹) ∈ On → ∀𝑧 ∈ suc dom recs(𝐹)(𝐶𝑧) = (𝐹‘(𝐶𝑧)))
12 fveq2 6649 . . . . . . 7 (𝑧 = 𝑦 → (𝐶𝑧) = (𝐶𝑦))
13 reseq2 5817 . . . . . . . 8 (𝑧 = 𝑦 → (𝐶𝑧) = (𝐶𝑦))
1413fveq2d 6653 . . . . . . 7 (𝑧 = 𝑦 → (𝐹‘(𝐶𝑧)) = (𝐹‘(𝐶𝑦)))
1512, 14eqeq12d 2817 . . . . . 6 (𝑧 = 𝑦 → ((𝐶𝑧) = (𝐹‘(𝐶𝑧)) ↔ (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
1615cbvralvw 3399 . . . . 5 (∀𝑧 ∈ suc dom recs(𝐹)(𝐶𝑧) = (𝐹‘(𝐶𝑧)) ↔ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))
1711, 16sylib 221 . . . 4 (dom recs(𝐹) ∈ On → ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))
18 fneq2 6419 . . . . . 6 (𝑥 = suc dom recs(𝐹) → (𝐶 Fn 𝑥𝐶 Fn suc dom recs(𝐹)))
19 raleq 3361 . . . . . 6 (𝑥 = suc dom recs(𝐹) → (∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)) ↔ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦))))
2018, 19anbi12d 633 . . . . 5 (𝑥 = suc dom recs(𝐹) → ((𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))) ↔ (𝐶 Fn suc dom recs(𝐹) ∧ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
2120rspcev 3574 . . . 4 ((suc dom recs(𝐹) ∈ On ∧ (𝐶 Fn suc dom recs(𝐹) ∧ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))) → ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
227, 9, 17, 21syl12anc 835 . . 3 (dom recs(𝐹) ∈ On → ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
236, 22syl 17 . 2 (recs(𝐹) ∈ V → ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
24 snex 5300 . . . . 5 {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V
25 unexg 7456 . . . . 5 ((recs(𝐹) ∈ V ∧ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V) → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
2624, 25mpan2 690 . . . 4 (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
278, 26eqeltrid 2897 . . 3 (recs(𝐹) ∈ V → 𝐶 ∈ V)
28 fneq1 6418 . . . . . 6 (𝑓 = 𝐶 → (𝑓 Fn 𝑥𝐶 Fn 𝑥))
29 fveq1 6648 . . . . . . . 8 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
30 reseq1 5816 . . . . . . . . 9 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
3130fveq2d 6653 . . . . . . . 8 (𝑓 = 𝐶 → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝐶𝑦)))
3229, 31eqeq12d 2817 . . . . . . 7 (𝑓 = 𝐶 → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
3332ralbidv 3165 . . . . . 6 (𝑓 = 𝐶 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
3428, 33anbi12d 633 . . . . 5 (𝑓 = 𝐶 → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3534rexbidv 3259 . . . 4 (𝑓 = 𝐶 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3635, 1elab2g 3619 . . 3 (𝐶 ∈ V → (𝐶𝐴 ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3727, 36syl 17 . 2 (recs(𝐹) ∈ V → (𝐶𝐴 ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3823, 37mpbird 260 1 (recs(𝐹) ∈ V → 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  {cab 2779  wral 3109  wrex 3110  Vcvv 3444  cun 3882  {csn 4528  cop 4534  dom cdm 5523  cres 5525  Ord word 6162  Oncon0 6163  suc csuc 6165   Fn wfn 6323  cfv 6328  recscrecs 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-wrecs 7934  df-recs 7995
This theorem is referenced by:  tfrlem13  8013
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