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Theorem csbwrecsg 33303
Description: Move class substitution in and out of the well-founded recursive function generator . (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csbwrecsg (𝐴𝑉𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹))

Proof of Theorem csbwrecsg
Dummy variables 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbuni 4498 . . 3 𝐴 / 𝑥 {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = 𝐴 / 𝑥{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
2 csbab 4041 . . . . 5 𝐴 / 𝑥{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓[𝐴 / 𝑥]𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
3 sbcex2 3519 . . . . . . 7 ([𝐴 / 𝑥]𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ∃𝑧[𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
4 sbc3an 3527 . . . . . . . . 9 ([𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ([𝐴 / 𝑥]𝑓 Fn 𝑧[𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ [𝐴 / 𝑥]𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
5 sbcg 3536 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]𝑓 Fn 𝑧𝑓 Fn 𝑧))
6 sbcan 3511 . . . . . . . . . . 11 ([𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ ([𝐴 / 𝑥]𝑧𝐷[𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
7 sbcssg 4118 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐷𝐴 / 𝑥𝑧𝐴 / 𝑥𝐷))
8 csbconstg 3579 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
98sseq1d 3665 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝐷𝑧𝐴 / 𝑥𝐷))
107, 9bitrd 268 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐷𝑧𝐴 / 𝑥𝐷))
11 sbcralg 3546 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
12 sbcssg 4118 . . . . . . . . . . . . . . 15 (𝐴𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝐴 / 𝑥𝑧))
138sseq2d 3666 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝐴 / 𝑥𝑧𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
14 csbpredg 33302 . . . . . . . . . . . . . . . . 17 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑦))
15 csbconstg 3579 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
16 predeq3 5722 . . . . . . . . . . . . . . . . . 18 (𝐴 / 𝑥𝑦 = 𝑦 → Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))
1715, 16syl 17 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))
1814, 17eqtrd 2685 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))
1918sseq1d 3665 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2012, 13, 193bitrd 294 . . . . . . . . . . . . . 14 (𝐴𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2120ralbidv 3015 . . . . . . . . . . . . 13 (𝐴𝑉 → (∀𝑦𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2211, 21bitrd 268 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2310, 22anbi12d 747 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥]𝑧𝐷[𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧)))
246, 23syl5bb 272 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧)))
25 sbcralg 3546 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦𝑧 [𝐴 / 𝑥](𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
26 sbceqg 4017 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥](𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ 𝐴 / 𝑥(𝑓𝑦) = 𝐴 / 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
27 csbconstg 3579 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥(𝑓𝑦) = (𝑓𝑦))
28 csbfv12 6269 . . . . . . . . . . . . . . 15 𝐴 / 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))
29 csbres 5431 . . . . . . . . . . . . . . . . 17 𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (𝐴 / 𝑥𝑓𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦))
30 csbconstg 3579 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑓 = 𝑓)
3130, 18reseq12d 5429 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → (𝐴 / 𝑥𝑓𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))
3229, 31syl5eq 2697 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))
3332fveq2d 6233 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))
3428, 33syl5eq 2697 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))
3527, 34eqeq12d 2666 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝐴 / 𝑥(𝑓𝑦) = 𝐴 / 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))))
3626, 35bitrd 268 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥](𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))))
3736ralbidv 3015 . . . . . . . . . . 11 (𝐴𝑉 → (∀𝑦𝑧 [𝐴 / 𝑥](𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦𝑧 (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))))
3825, 37bitrd 268 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦𝑧 (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))))
395, 24, 383anbi123d 1439 . . . . . . . . 9 (𝐴𝑉 → (([𝐴 / 𝑥]𝑓 Fn 𝑧[𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ [𝐴 / 𝑥]𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ (𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))))
404, 39syl5bb 272 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ (𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))))
4140exbidv 1890 . . . . . . 7 (𝐴𝑉 → (∃𝑧[𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))))
423, 41syl5bb 272 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))))
4342abbidv 2770 . . . . 5 (𝐴𝑉 → {𝑓[𝐴 / 𝑥]𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))})
442, 43syl5eq 2697 . . . 4 (𝐴𝑉𝐴 / 𝑥{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))})
4544unieqd 4478 . . 3 (𝐴𝑉 𝐴 / 𝑥{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))})
461, 45syl5eq 2697 . 2 (𝐴𝑉𝐴 / 𝑥 {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))})
47 df-wrecs 7452 . . 3 wrecs(𝑅, 𝐷, 𝐹) = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
4847csbeq2i 4026 . 2 𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = 𝐴 / 𝑥 {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
49 df-wrecs 7452 . 2 wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹) = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))}
5046, 48, 493eqtr4g 2710 1 (𝐴𝑉𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  [wsbc 3468  csb 3566  wss 3607   cuni 4468  cres 5145  Predcpred 5717   Fn wfn 5921  cfv 5926  wrecscwrecs 7451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-iota 5889  df-fv 5934  df-wrecs 7452
This theorem is referenced by:  csbrecsg  33304
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