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Theorem frecseq123 33016
Description: Equality theorem for 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 1129 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → 𝐴 = 𝐵)
21sseq2d 3996 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑥𝐴𝑥𝐵))
3 equid 2010 . . . . . . . . . . 11 𝑦 = 𝑦
4 predeq123 6142 . . . . . . . . . . 11 ((𝑅 = 𝑆𝐴 = 𝐵𝑦 = 𝑦) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
53, 4mp3an3 1441 . . . . . . . . . 10 ((𝑅 = 𝑆𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
653adant3 1124 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
76sseq1d 3995 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
87ralbidv 3194 . . . . . . 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 1130 . . . . . . . . . 10 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → 𝐹 = 𝐺)
1110oveqd 7162 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
126reseq2d 5846 . . . . . . . . . 10 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
1312oveq2d 7161 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))
1411, 13eqtrd 2853 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))
1514eqeq2d 2829 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → ((𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
1615ralbidv 3194 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
179, 163anbi23d 1430 . . . . 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 1913 . . . 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 2882 . . 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 4840 . 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 33015 . 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 33015 . 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 2878 1 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → frecs(𝑅, 𝐴, 𝐹) = frecs(𝑆, 𝐵, 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wex 1771  {cab 2796  wral 3135  wss 3933   cuni 4830  cres 5550  Predcpred 6140   Fn wfn 6343  cfv 6348  (class class class)co 7145  frecscfrecs 33014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fv 6356  df-ov 7148  df-frecs 33015
This theorem is referenced by: (None)
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