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Theorem wrecseq123OLD 8246
Description: Obsolete proof of wrecseq123 8245 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 7-Jun-2018.)
Assertion
Ref Expression
wrecseq123OLD ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Proof of Theorem wrecseq123OLD
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq2 3970 . . . . . . . 8 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
213ad2ant2 1134 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑥𝐴𝑥𝐵))
3 predeq1 6255 . . . . . . . . . . 11 (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐴, 𝑦))
4 predeq2 6256 . . . . . . . . . . 11 (𝐴 = 𝐵 → Pred(𝑆, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
53, 4sylan9eq 2796 . . . . . . . . . 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 3975 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
87ralbidv 3174 . . . . . . 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 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → 𝐹 = 𝐺)
115reseq2d 5937 . . . . . . . . . 10 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
12113adant3 1132 . . . . . . . . 9 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
1310, 12fveq12d 6849 . . . . . . . 8 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))
1413eqeq2d 2747 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → ((𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
1514ralbidv 3174 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
169, 153anbi23d 1439 . . . . 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 1924 . . . 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 2805 . . 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 4879 . 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 dfwrecsOLD 8244 . 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 dfwrecsOLD 8244 . 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 2801 1 ((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  {cab 2713  wral 3064  wss 3910   cuni 4865  cres 5635  Predcpred 6252   Fn wfn 6491  cfv 6496  wrecscwrecs 8242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243
This theorem is referenced by: (None)
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