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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 ∧ ∀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 7633 . . . . 5 Ord dom recs(𝐹)
32a1i 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))
63, 4, 5sylanbrc 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))⟩})
91, 8tfrlem10 7636 . . . 4 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
101, 8tfrlem11 7637 . . . . . 6 (dom recs(𝐹) ∈ On → (𝑧 ∈ suc dom recs(𝐹) → (𝐶𝑧) = (𝐹‘(𝐶𝑧))))
1110ralrimiv 3114 . . . . 5 (dom recs(𝐹) ∈ On → ∀𝑧 ∈ suc dom recs(𝐹)(𝐶𝑧) = (𝐹‘(𝐶𝑧)))
12 fveq2 6332 . . . . . . 7 (𝑧 = 𝑦 → (𝐶𝑧) = (𝐶𝑦))
13 reseq2 5529 . . . . . . . 8 (𝑧 = 𝑦 → (𝐶𝑧) = (𝐶𝑦))
1413fveq2d 6336 . . . . . . 7 (𝑧 = 𝑦 → (𝐹‘(𝐶𝑧)) = (𝐹‘(𝐶𝑦)))
1512, 14eqeq12d 2786 . . . . . 6 (𝑧 = 𝑦 → ((𝐶𝑧) = (𝐹‘(𝐶𝑧)) ↔ (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
1615cbvralv 3320 . . . . 5 (∀𝑧 ∈ suc dom recs(𝐹)(𝐶𝑧) = (𝐹‘(𝐶𝑧)) ↔ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))
1711, 16sylib 208 . . . 4 (dom recs(𝐹) ∈ On → ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))
18 fneq2 6120 . . . . . 6 (𝑥 = suc dom recs(𝐹) → (𝐶 Fn 𝑥𝐶 Fn suc dom recs(𝐹)))
19 raleq 3287 . . . . . 6 (𝑥 = suc dom recs(𝐹) → (∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)) ↔ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦))))
2018, 19anbi12d 616 . . . . 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 3460 . . . 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 1474 . . 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 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)
2624, 25mpan2 671 . . . 4 (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
278, 26syl5eqel 2854 . . 3 (recs(𝐹) ∈ V → 𝐶 ∈ V)
28 fneq1 6119 . . . . . 6 (𝑓 = 𝐶 → (𝑓 Fn 𝑥𝐶 Fn 𝑥))
29 fveq1 6331 . . . . . . . 8 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
30 reseq1 5528 . . . . . . . . 9 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
3130fveq2d 6336 . . . . . . . 8 (𝑓 = 𝐶 → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝐶𝑦)))
3229, 31eqeq12d 2786 . . . . . . 7 (𝑓 = 𝐶 → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
3332ralbidv 3135 . . . . . 6 (𝑓 = 𝐶 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
3428, 33anbi12d 616 . . . . 5 (𝑓 = 𝐶 → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3534rexbidv 3200 . . . 4 (𝑓 = 𝐶 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3635, 1elab2g 3504 . . 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 247 1 (recs(𝐹) ∈ V → 𝐶𝐴)
Colors of variables: wff setvar class
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
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