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Theorem csbwrecsg 32349
 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 4402 . . 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 3960 . . . . 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 3453 . . . . . . 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 3461 . . . . . . . . 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 3470 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]𝑓 Fn 𝑧𝑓 Fn 𝑧))
6 sbcan 3445 . . . . . . . . . . 11 ([𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ ([𝐴 / 𝑥]𝑧𝐷[𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
7 sbcssg 4035 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐷𝐴 / 𝑥𝑧𝐴 / 𝑥𝐷))
8 csbconstg 3512 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
98sseq1d 3595 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝐷𝑧𝐴 / 𝑥𝐷))
107, 9bitrd 267 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐷𝑧𝐴 / 𝑥𝐷))
11 sbcralg 3480 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
12 sbcssg 4035 . . . . . . . . . . . . . . 15 (𝐴𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝐴 / 𝑥𝑧))
138sseq2d 3596 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝐴 / 𝑥𝑧𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
14 csbpredg 32348 . . . . . . . . . . . . . . . . 17 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑦))
15 csbconstg 3512 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
16 predeq3 5601 . . . . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))
1918sseq1d 3595 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2012, 13, 193bitrd 293 . . . . . . . . . . . . . 14 (𝐴𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2120ralbidv 2969 . . . . . . . . . . . . 13 (𝐴𝑉 → (∀𝑦𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2211, 21bitrd 267 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧))
2310, 22anbi12d 743 . . . . . . . . . . 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 271 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝐷 ∧ ∀𝑦𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ (𝑧𝐴 / 𝑥𝐷 ∧ ∀𝑦𝑧 Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦) ⊆ 𝑧)))
25 sbcralg 3480 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦𝑧 [𝐴 / 𝑥](𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
26 sbceqg 3936 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥](𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ 𝐴 / 𝑥(𝑓𝑦) = 𝐴 / 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
27 csbconstg 3512 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥(𝑓𝑦) = (𝑓𝑦))
28 csbfv12 6141 . . . . . . . . . . . . . . 15 𝐴 / 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))
29 csbres 5320 . . . . . . . . . . . . . . . . 17 𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (𝐴 / 𝑥𝑓𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦))
30 csbconstg 3512 . . . . . . . . . . . . . . . . . 18 (𝐴𝑉𝐴 / 𝑥𝑓 = 𝑓)
3130, 18reseq12d 5318 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → (𝐴 / 𝑥𝑓𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))
3229, 31syl5eq 2656 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))
3332fveq2d 6107 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))
3428, 33syl5eq 2656 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦))))
3527, 34eqeq12d 2625 . . . . . . . . . . . . 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 267 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥](𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ (𝑓𝑦) = (𝐴 / 𝑥𝐹‘(𝑓 ↾ Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝑦)))))
3736ralbidv 2969 . . . . . . . . . . 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 267 . . . . . . . . . 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 1391 . . . . . . . . 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 271 . . . . . . . 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 1837 . . . . . . 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 271 . . . . . 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 2728 . . . . 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 2656 . . . 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 4382 . . 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 2656 . 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 7294 . . 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 3945 . 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 7294 . 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 2669 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 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  [wsbc 3402  ⦋csb 3499   ⊆ wss 3540  ∪ cuni 4372   ↾ cres 5040  Predcpred 5596   Fn wfn 5799  ‘cfv 5804  wrecscwrecs 7293 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-wrecs 7294 This theorem is referenced by:  csbrecsg  32350
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