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	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))

Proof of Theorem csbwrecsg

Dummy variables 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbuni 4402	. . 3 ⊢ ⦋𝐴 / 𝑥⦌∪ {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = ∪ ⦋𝐴 / 𝑥⦌{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
2		csbab 3960	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓 ∣ [𝐴 / 𝑥]∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
3		sbcex2 3453	. . . . . . 7 ⊢ ([𝐴 / 𝑥]∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ∃𝑧[𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
4		sbc3an 3461	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ([𝐴 / 𝑥]𝑓 Fn 𝑧 ∧ [𝐴 / 𝑥](𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ [𝐴 / 𝑥]∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
5		sbcg 3470	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑓 Fn 𝑧 ↔ 𝑓 Fn 𝑧))
6		sbcan 3445	. . . . . . . . . . 11 ⊢ ([𝐴 / 𝑥](𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ ([𝐴 / 𝑥]𝑧 ⊆ 𝐷 ∧ [𝐴 / 𝑥]∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
7		sbcssg 4035	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷))
8		csbconstg 3512	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
9	8	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ↔ 𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷))
10	7, 9	bitrd 267	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ⊆ 𝐷 ↔ 𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷))
11		sbcralg 3480	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
12		sbcssg 4035	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦) ⊆ ⦋𝐴 / 𝑥⦌𝑧))
13	8	sseq2d 3596	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦) ⊆ ⦋𝐴 / 𝑥⦌𝑧 ↔ ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
14		csbpredg 32348	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑦))
15		csbconstg 3512	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
16		predeq3 5601	. . . . . . . . . . . . . . . . . 18 ⊢ (⦋𝐴 / 𝑥⦌𝑦 = 𝑦 → Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑦) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))
17	15, 16	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ 𝑉 → Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑦) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))
18	14, 17	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))
19	18	sseq1d 3595	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧))
20	12, 13, 19	3bitrd 293	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧))
21	20	ralbidv 2969	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ 𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧))
22	11, 21	bitrd 267	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧))
23	10, 22	anbi12d 743	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑧 ⊆ 𝐷 ∧ [𝐴 / 𝑥]∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ (𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧)))
24	6, 23	syl5bb 271	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ↔ (𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧)))
25		sbcralg 3480	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦 ∈ 𝑧 [𝐴 / 𝑥](𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
26		sbceqg 3936	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ⦋𝐴 / 𝑥⦌(𝑓‘𝑦) = ⦋𝐴 / 𝑥⦌(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
27		csbconstg 3512	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑓‘𝑦) = (𝑓‘𝑦))
28		csbfv12 6141	. . . . . . . . . . . . . . 15 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))
29		csbres 5320	. . . . . . . . . . . . . . . . 17 ⊢ ⦋𝐴 / 𝑥⦌(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (⦋𝐴 / 𝑥⦌𝑓 ↾ ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦))
30		csbconstg 3512	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑓 = 𝑓)
31	30, 18	reseq12d 5318	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑓 ↾ ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))
32	29, 31	syl5eq 2656	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))
33	32	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))
34	28, 33	syl5eq 2656	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))
35	27, 34	eqeq12d 2625	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌(𝑓‘𝑦) = ⦋𝐴 / 𝑥⦌(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))))
36	26, 35	bitrd 267	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))))
37	36	ralbidv 2969	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ 𝑧 [𝐴 / 𝑥](𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))))
38	25, 37	bitrd 267	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))))
39	5, 24, 38	3anbi123d 1391	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑓 Fn 𝑧 ∧ [𝐴 / 𝑥](𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ [𝐴 / 𝑥]∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ (𝑓 Fn 𝑧 ∧ (𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))))
40	4, 39	syl5bb 271	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ (𝑓 Fn 𝑧 ∧ (𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))))
41	40	exbidv 1837	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (∃𝑧[𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))))
42	3, 41	syl5bb 271	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))))
43	42	abbidv 2728	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ [𝐴 / 𝑥]∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))})
44	2, 43	syl5eq 2656	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))})
45	44	unieqd 4382	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ ⦋𝐴 / 𝑥⦌{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = ∪ {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))})
46	1, 45	syl5eq 2656	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))} = ∪ {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))})
47		df-wrecs 7294	. . 3 ⊢ wrecs(𝑅, 𝐷, 𝐹) = ∪ {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
48	47	csbeq2i 3945	. 2 ⊢ ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = ⦋𝐴 / 𝑥⦌∪ {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
49		df-wrecs 7294	. 2 ⊢ wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹) = ∪ {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))}
50	46, 48, 49	3eqtr4g 2669	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))

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