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Theorem wrecseq123 7799
Description: General equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
Assertion
Ref Expression
wrecseq123 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Proof of Theorem wrecseq123
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq2 3914 . . . . . . . 8 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
213ad2ant2 1127 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑥𝐴𝑥𝐵))
3 predeq1 6025 . . . . . . . . . . 11 (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐴, 𝑦))
4 predeq2 6026 . . . . . . . . . . 11 (𝐴 = 𝐵 → Pred(𝑆, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
53, 4sylan9eq 2851 . . . . . . . . . 10 ((𝑅 = 𝑆𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
653adant3 1125 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
76sseq1d 3919 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
87ralbidv 3164 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
92, 8anbi12d 630 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥)))
10 simp3 1131 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → 𝐹 = 𝐺)
115reseq2d 5734 . . . . . . . . . 10 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
12113adant3 1125 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
1310, 12fveq12d 6545 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))
1413eqeq2d 2805 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → ((𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
1514ralbidv 3164 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
169, 153anbi23d 1431 . . . . 5 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))))
1716exbidv 1899 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))))
1817abbidv 2860 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))})
1918unieqd 4755 . 2 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))})
20 df-wrecs 7798 . 2 wrecs(𝑅, 𝐴, 𝐹) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
21 df-wrecs 7798 . 2 wrecs(𝑆, 𝐵, 𝐺) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))}
2219, 20, 213eqtr4g 2856 1 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wex 1761  {cab 2775  wral 3105  wss 3859   cuni 4745  cres 5445  Predcpred 6022   Fn wfn 6220  cfv 6225  wrecscwrecs 7797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-xp 5449  df-cnv 5451  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-iota 6189  df-fv 6233  df-wrecs 7798
This theorem is referenced by:  wrecseq1  7801  wrecseq2  7802  wrecseq3  7803
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