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Theorem csbfrecsg 8270
Description: Move class substitution in and out of the well-founded recursive function generator. (Contributed by Scott Fenton, 18-Nov-2024.)
Assertion
Ref Expression
csbfrecsg (𝐴𝑉𝐴 / 𝑥frecs(𝑅, 𝐷, 𝐹) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹))

Proof of Theorem csbfrecsg
Dummy variables 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbuni 4933 . . 3 𝐴 / 𝑥 {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = 𝐴 / 𝑥{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
2 csbab 4432 . . . . 5 𝐴 / 𝑥{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓[𝐴 / 𝑥]𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
3 sbcex2 3837 . . . . . . 7 ([𝐴 / 𝑥]𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ∃𝑧[𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
4 sbc3an 3842 . . . . . . . . 9 ([𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ([𝐴 / 𝑥]𝑓 Fn 𝑧[𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ [𝐴 / 𝑥]𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
5 sbcg 3851 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]𝑓 Fn 𝑧𝑓 Fn 𝑧))
6 sbcan 3824 . . . . . . . . . . 11 ([𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ ([𝐴 / 𝑥]𝑧𝐷[𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
7 sbcssg 4518 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐷𝐴 / 𝑥𝑧𝐴 / 𝑥𝐷))
8 csbconstg 3907 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
98sseq1d 4008 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝐷𝑧𝐴 / 𝑥𝐷))
107, 9bitrd 279 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐷𝑧𝐴 / 𝑥𝐷))
11 sbcralg 3863 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
12 sbcssg 4518 . . . . . . . . . . . . . . 15 (𝐴𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝐴 / 𝑥𝑧))
13 csbpredg 6300 . . . . . . . . . . . . . . . . 17 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑦))
14 csbconstg 3907 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
15 predeq3 6298 . . . . . . . . . . . . . . . . . 18 (𝐴 / 𝑥𝑦 = 𝑦 → Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))
1614, 15syl 17 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))
1713, 16eqtrd 2766 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))
1817, 8sseq12d 4010 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝐴 / 𝑥𝑧 ↔ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
1912, 18bitrd 279 . . . . . . . . . . . . . 14 (𝐴𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2019ralbidv 3171 . . . . . . . . . . . . 13 (𝐴𝑉 → (∀𝑦𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2111, 20bitrd 279 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2210, 21anbi12d 630 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥]𝑧𝐷[𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧)))
236, 22bitrid 283 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧)))
24 sbcralg 3863 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦𝑧 [𝐴 / 𝑥](𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
25 sbceqg 4404 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥](𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ 𝐴 / 𝑥(𝑓𝑦) = 𝐴 / 𝑥(𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
26 csbconstg 3907 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥(𝑓𝑦) = (𝑓𝑦))
27 csbov123 7447 . . . . . . . . . . . . . . 15 𝐴 / 𝑥(𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝐴 / 𝑥𝑦𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))
28 csbres 5978 . . . . . . . . . . . . . . . . 17 𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (𝐴 / 𝑥𝑓𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦))
29 csbconstg 3907 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑓 = 𝑓)
3029, 17reseq12d 5976 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → (𝐴 / 𝑥𝑓𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))
3128, 30eqtrid 2778 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))
3214, 31oveq12d 7423 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥𝑦𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))
3327, 32eqtrid 2778 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥(𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))
3426, 33eqeq12d 2742 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝐴 / 𝑥(𝑓𝑦) = 𝐴 / 𝑥(𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))))
3525, 34bitrd 279 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥](𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))))
3635ralbidv 3171 . . . . . . . . . . 11 (𝐴𝑉 → (∀𝑦𝑧 [𝐴 / 𝑥](𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))))
3724, 36bitrd 279 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))))
385, 23, 373anbi123d 1432 . . . . . . . . 9 (𝐴𝑉 → (([𝐴 / 𝑥]𝑓 Fn 𝑧[𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ [𝐴 / 𝑥]𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ (𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))))
394, 38bitrid 283 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ (𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))))
4039exbidv 1916 . . . . . . 7 (𝐴𝑉 → (∃𝑧[𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))))
413, 40bitrid 283 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))))
4241abbidv 2795 . . . . 5 (𝐴𝑉 → {𝑓[𝐴 / 𝑥]𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))})
432, 42eqtrid 2778 . . . 4 (𝐴𝑉𝐴 / 𝑥{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))})
4443unieqd 4915 . . 3 (𝐴𝑉 𝐴 / 𝑥{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))})
451, 44eqtrid 2778 . 2 (𝐴𝑉𝐴 / 𝑥 {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))})
46 df-frecs 8267 . . 3 frecs(𝑅, 𝐷, 𝐹) = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
4746csbeq2i 3896 . 2 𝐴 / 𝑥frecs(𝑅, 𝐷, 𝐹) = 𝐴 / 𝑥 {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
48 df-frecs 8267 . 2 frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹) = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦𝑧 (𝑓𝑦) = (𝑦𝐴 / 𝑥𝐹(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))}
4945, 47, 483eqtr4g 2791 1 (𝐴𝑉𝐴 / 𝑥frecs(𝑅, 𝐷, 𝐹) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2703  wral 3055  [wsbc 3772  csb 3888  wss 3943   cuni 4902  cres 5671  Predcpred 6293   Fn wfn 6532  cfv 6537  (class class class)co 7405  frecscfrecs 8266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-iota 6489  df-fv 6545  df-ov 7408  df-frecs 8267
This theorem is referenced by:  csbwrecsg  8307
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