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Theorem csbwrecsg 34047
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 4740 . . 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 4273 . . . . 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 3738 . . . . . . 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 3742 . . . . . . . . 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 3751 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]𝑓 Fn 𝑧𝑓 Fn 𝑧))
6 sbcan 3726 . . . . . . . . . . 11 ([𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ ([𝐴 / 𝑥]𝑧𝐷[𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
7 sbcssg 4346 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐷𝐴 / 𝑥𝑧𝐴 / 𝑥𝐷))
8 csbconstg 3800 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
98sseq1d 3889 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝐷𝑧𝐴 / 𝑥𝐷))
107, 9bitrd 271 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐷𝑧𝐴 / 𝑥𝐷))
11 sbcralg 3761 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
12 sbcssg 4346 . . . . . . . . . . . . . . 15 (𝐴𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝐴 / 𝑥𝑧))
138sseq2d 3890 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝐴 / 𝑥𝑧𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
14 csbpredg 34046 . . . . . . . . . . . . . . . . 17 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑦))
15 csbconstg 3800 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
16 predeq3 5990 . . . . . . . . . . . . . . . . . 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 2815 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))
1918sseq1d 3889 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2012, 13, 193bitrd 297 . . . . . . . . . . . . . 14 (𝐴𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2120ralbidv 3148 . . . . . . . . . . . . 13 (𝐴𝑉 → (∀𝑦𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2211, 21bitrd 271 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2310, 22anbi12d 621 . . . . . . . . . . 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 275 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧)))
25 sbcralg 3761 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦𝑧 [𝐴 / 𝑥](𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
26 sbceqg 4247 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥](𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ 𝐴 / 𝑥(𝑓𝑦) = 𝐴 / 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
27 csbconstg 3800 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥(𝑓𝑦) = (𝑓𝑦))
28 csbfv12 6543 . . . . . . . . . . . . . . 15 𝐴 / 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))
29 csbres 5698 . . . . . . . . . . . . . . . . 17 𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (𝐴 / 𝑥𝑓𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦))
30 csbconstg 3800 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑓 = 𝑓)
3130, 18reseq12d 5696 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → (𝐴 / 𝑥𝑓𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))
3229, 31syl5eq 2827 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))
3332fveq2d 6503 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))
3428, 33syl5eq 2827 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))
3527, 34eqeq12d 2794 . . . . . . . . . . . . 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 271 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥](𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))))
3736ralbidv 3148 . . . . . . . . . . 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 271 . . . . . . . . . 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 1415 . . . . . . . . 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 275 . . . . . . . 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 1880 . . . . . . 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 275 . . . . . 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 2844 . . . . 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 2827 . . . 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 4722 . . 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 2827 . 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 7750 . . 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 4257 . 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 7750 . 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 2840 1 (𝐴𝑉𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wex 1742  wcel 2050  {cab 2759  wral 3089  [wsbc 3682  csb 3787  wss 3830   cuni 4712  cres 5409  Predcpred 5985   Fn wfn 6183  cfv 6188  wrecscwrecs 7749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-xp 5413  df-cnv 5415  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-iota 6152  df-fv 6196  df-wrecs 7750
This theorem is referenced by:  csbrecsg  34048
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