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Theorem tfrlem12 7431
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 7426 . . . . 5 Ord dom recs(𝐹)
32a1i 11 . . . 4 (recs(𝐹) ∈ V → Ord dom recs(𝐹))
4 dmexg 7045 . . . 4 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
5 elon2 5696 . . . 4 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
63, 4, 5sylanbrc 697 . . 3 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
7 suceloni 6961 . . . 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 7429 . . . 4 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
101, 8tfrlem11 7430 . . . . . 6 (dom recs(𝐹) ∈ On → (𝑧 ∈ suc dom recs(𝐹) → (𝐶𝑧) = (𝐹‘(𝐶𝑧))))
1110ralrimiv 2964 . . . . 5 (dom recs(𝐹) ∈ On → ∀𝑧 ∈ suc dom recs(𝐹)(𝐶𝑧) = (𝐹‘(𝐶𝑧)))
12 fveq2 6150 . . . . . . 7 (𝑧 = 𝑦 → (𝐶𝑧) = (𝐶𝑦))
13 reseq2 5355 . . . . . . . 8 (𝑧 = 𝑦 → (𝐶𝑧) = (𝐶𝑦))
1413fveq2d 6154 . . . . . . 7 (𝑧 = 𝑦 → (𝐹‘(𝐶𝑧)) = (𝐹‘(𝐶𝑦)))
1512, 14eqeq12d 2641 . . . . . 6 (𝑧 = 𝑦 → ((𝐶𝑧) = (𝐹‘(𝐶𝑧)) ↔ (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
1615cbvralv 3164 . . . . 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 5940 . . . . . 6 (𝑥 = suc dom recs(𝐹) → (𝐶 Fn 𝑥𝐶 Fn suc dom recs(𝐹)))
19 raleq 3132 . . . . . 6 (𝑥 = suc dom recs(𝐹) → (∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)) ↔ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦))))
2018, 19anbi12d 746 . . . . 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 3300 . . . 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 1321 . . 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 4874 . . . . 5 {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V
25 unexg 6913 . . . . 5 ((recs(𝐹) ∈ V ∧ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V) → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
2624, 25mpan2 706 . . . 4 (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
278, 26syl5eqel 2708 . . 3 (recs(𝐹) ∈ V → 𝐶 ∈ V)
28 fneq1 5939 . . . . . 6 (𝑓 = 𝐶 → (𝑓 Fn 𝑥𝐶 Fn 𝑥))
29 fveq1 6149 . . . . . . . 8 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
30 reseq1 5354 . . . . . . . . 9 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
3130fveq2d 6154 . . . . . . . 8 (𝑓 = 𝐶 → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝐶𝑦)))
3229, 31eqeq12d 2641 . . . . . . 7 (𝑓 = 𝐶 → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
3332ralbidv 2985 . . . . . 6 (𝑓 = 𝐶 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
3428, 33anbi12d 746 . . . . 5 (𝑓 = 𝐶 → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3534rexbidv 3050 . . . 4 (𝑓 = 𝐶 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3635, 1elab2g 3341 . . 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 384   = wceq 1480  wcel 1992  {cab 2612  wral 2912  wrex 2913  Vcvv 3191  cun 3558  {csn 4153  cop 4159  dom cdm 5079  cres 5081  Ord word 5684  Oncon0 5685  suc csuc 5687   Fn wfn 5845  cfv 5850  recscrecs 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-fv 5858  df-wrecs 7353  df-recs 7414
This theorem is referenced by:  tfrlem13  7432
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