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Theorem wrecseq123 7295
 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 3590 . . . . . . . 8 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
213ad2ant2 1076 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑥𝐴𝑥𝐵))
3 predeq1 5599 . . . . . . . . . . 11 (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐴, 𝑦))
4 predeq2 5600 . . . . . . . . . . 11 (𝐴 = 𝐵 → Pred(𝑆, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
53, 4sylan9eq 2664 . . . . . . . . . 10 ((𝑅 = 𝑆𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
653adant3 1074 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
76sseq1d 3595 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
87ralbidv 2969 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
92, 8anbi12d 743 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥)))
10 simp3 1056 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → 𝐹 = 𝐺)
115reseq2d 5317 . . . . . . . . . 10 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
12113adant3 1074 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
1310, 12fveq12d 6109 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))
1413eqeq2d 2620 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → ((𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
1514ralbidv 2969 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
169, 153anbi23d 1394 . . . . 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 1837 . . . 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 2728 . . 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 4382 . 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 7294 . 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 7294 . 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 2669 1 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695  {cab 2596  ∀wral 2896   ⊆ wss 3540  ∪ cuni 4372   ↾ cres 5040  Predcpred 5596   Fn wfn 5799  ‘cfv 5804  wrecscwrecs 7293 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-wrecs 7294 This theorem is referenced by:  wrecseq1  7297  wrecseq2  7298  wrecseq3  7299
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