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Theorem tfrlem12 7349
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 7344 . . . . 5 Ord dom recs(𝐹)
32a1i 11 . . . 4 (recs(𝐹) ∈ V → Ord dom recs(𝐹))
4 dmexg 6966 . . . 4 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
5 elon2 5637 . . . 4 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
63, 4, 5sylanbrc 694 . . 3 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
7 suceloni 6882 . . . 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 7347 . . . 4 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
101, 8tfrlem11 7348 . . . . . 6 (dom recs(𝐹) ∈ On → (𝑧 ∈ suc dom recs(𝐹) → (𝐶𝑧) = (𝐹‘(𝐶𝑧))))
1110ralrimiv 2947 . . . . 5 (dom recs(𝐹) ∈ On → ∀𝑧 ∈ suc dom recs(𝐹)(𝐶𝑧) = (𝐹‘(𝐶𝑧)))
12 fveq2 6088 . . . . . . 7 (𝑧 = 𝑦 → (𝐶𝑧) = (𝐶𝑦))
13 reseq2 5299 . . . . . . . 8 (𝑧 = 𝑦 → (𝐶𝑧) = (𝐶𝑦))
1413fveq2d 6092 . . . . . . 7 (𝑧 = 𝑦 → (𝐹‘(𝐶𝑧)) = (𝐹‘(𝐶𝑦)))
1512, 14eqeq12d 2624 . . . . . 6 (𝑧 = 𝑦 → ((𝐶𝑧) = (𝐹‘(𝐶𝑧)) ↔ (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
1615cbvralv 3146 . . . . 5 (∀𝑧 ∈ suc dom recs(𝐹)(𝐶𝑧) = (𝐹‘(𝐶𝑧)) ↔ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))
1711, 16sylib 206 . . . 4 (dom recs(𝐹) ∈ On → ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))
18 fneq2 5880 . . . . . 6 (𝑥 = suc dom recs(𝐹) → (𝐶 Fn 𝑥𝐶 Fn suc dom recs(𝐹)))
19 raleq 3114 . . . . . 6 (𝑥 = suc dom recs(𝐹) → (∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)) ↔ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦))))
2018, 19anbi12d 742 . . . . 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 3281 . . . 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 1315 . . 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 4830 . . . . 5 {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V
25 unexg 6834 . . . . 5 ((recs(𝐹) ∈ V ∧ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V) → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
2624, 25mpan2 702 . . . 4 (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
278, 26syl5eqel 2691 . . 3 (recs(𝐹) ∈ V → 𝐶 ∈ V)
28 fneq1 5879 . . . . . 6 (𝑓 = 𝐶 → (𝑓 Fn 𝑥𝐶 Fn 𝑥))
29 fveq1 6087 . . . . . . . 8 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
30 reseq1 5298 . . . . . . . . 9 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
3130fveq2d 6092 . . . . . . . 8 (𝑓 = 𝐶 → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝐶𝑦)))
3229, 31eqeq12d 2624 . . . . . . 7 (𝑓 = 𝐶 → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
3332ralbidv 2968 . . . . . 6 (𝑓 = 𝐶 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
3428, 33anbi12d 742 . . . . 5 (𝑓 = 𝐶 → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3534rexbidv 3033 . . . 4 (𝑓 = 𝐶 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3635, 1elab2g 3321 . . 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 245 1 (recs(𝐹) ∈ V → 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {cab 2595  wral 2895  wrex 2896  Vcvv 3172  cun 3537  {csn 4124  cop 4130  dom cdm 5028  cres 5030  Ord word 5625  Oncon0 5626  suc csuc 5628   Fn wfn 5785  cfv 5790  recscrecs 7331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798  df-wrecs 7271  df-recs 7332
This theorem is referenced by:  tfrlem13  7350
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