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Theorem frecseq123 8323
Description: Equality theorem for the well-founded recursion generator. (Contributed by Scott Fenton, 23-Dec-2021.)
Assertion
Ref Expression
frecseq123 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → frecs(𝑅, 𝐴, 𝐹) = frecs(𝑆, 𝐵, 𝐺))

Proof of Theorem frecseq123
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1137 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → 𝐴 = 𝐵)
21sseq2d 4041 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑥𝐴𝑥𝐵))
3 equid 2011 . . . . . . . . . . 11 𝑦 = 𝑦
4 predeq123 6333 . . . . . . . . . . 11 ((𝑅 = 𝑆𝐴 = 𝐵𝑦 = 𝑦) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
53, 4mp3an3 1450 . . . . . . . . . 10 ((𝑅 = 𝑆𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
653adant3 1132 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
76sseq1d 4040 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
87ralbidv 3184 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
92, 8anbi12d 631 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥)))
10 simp3 1138 . . . . . . . . . 10 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → 𝐹 = 𝐺)
1110oveqd 7465 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
126reseq2d 6009 . . . . . . . . . 10 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
1312oveq2d 7464 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))
1411, 13eqtrd 2780 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))
1514eqeq2d 2751 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → ((𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
1615ralbidv 3184 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
179, 163anbi23d 1439 . . . . 5 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))))
1817exbidv 1920 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))))
1918abbidv 2811 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))})
2019unieqd 4944 . 2 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))})
21 df-frecs 8322 . 2 frecs(𝑅, 𝐴, 𝐹) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
22 df-frecs 8322 . 2 frecs(𝑆, 𝐵, 𝐺) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐵 ∧ ∀𝑦𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))}
2320, 21, 223eqtr4g 2805 1 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → frecs(𝑅, 𝐴, 𝐹) = frecs(𝑆, 𝐵, 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1537  wex 1777  {cab 2717  wral 3067  wss 3976   cuni 4931  cres 5702  Predcpred 6331   Fn wfn 6568  cfv 6573  (class class class)co 7448  frecscfrecs 8321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fv 6581  df-ov 7451  df-frecs 8322
This theorem is referenced by:  wrecseq123  8355  csbwrecsg  8362
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