Theorem csbfrecsg 8333

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

Proof of Theorem csbfrecsg

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

Step	Hyp	Ref	Expression
1		csbuni 4901	. . 3 ⊢ ⦋𝐴 / 𝑥⦌∪ {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(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) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
2		csbab 4401	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(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) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
3		sbcex2 3802	. . . . . . 7 ⊢ ([𝐴 / 𝑥]∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))) ↔ ∃𝑧[𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
4		sbc3an 3806	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(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) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
5		sbcg 3814	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑓 Fn 𝑧 ↔ 𝑓 Fn 𝑧))
6		sbcan 3791	. . . . . . . . . . 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 4480	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷))
8		csbconstg 3870	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
9	8	sseq1d 3944	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ↔ 𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷))
10	7, 9	bitrd 282	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ⊆ 𝐷 ↔ 𝑧 ⊆ ⦋𝐴 / 𝑥⦌𝐷))
11		sbcralg 3825	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧))
12		sbcssg 4480	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦) ⊆ ⦋𝐴 / 𝑥⦌𝑧))
13		csbpredg 6331	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑦))
14		csbconstg 3870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
15		predeq3 6329	. . . . . . . . . . . . . . . . . 18 ⊢ (⦋𝐴 / 𝑥⦌𝑦 = 𝑦 → Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑦) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))
16	14, 15	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ 𝑉 → Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑦) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))
17	13, 16	eqtrd 2798	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))
18	17, 8	sseq12d 3946	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦) ⊆ ⦋𝐴 / 𝑥⦌𝑧 ↔ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧))
19	12, 18	bitrd 282	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧))
20	19	ralbidv 3189	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ 𝑧 [𝐴 / 𝑥]Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧))
21	11, 20	bitrd 282	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑧 Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦) ⊆ 𝑧))
22	10, 21	anbi12d 644	. . . . . . . . . . 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)))
23	6, 22	bitrid 286	. . . . . . . . . 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)))
24		sbcralg 3825	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦 ∈ 𝑧 [𝐴 / 𝑥](𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
25		sbceqg 4373	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ⦋𝐴 / 𝑥⦌(𝑓‘𝑦) = ⦋𝐴 / 𝑥⦌(𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))))
26		csbconstg 3870	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑓‘𝑦) = (𝑓‘𝑦))
27		csbov123 7490	. . . . . . . . . . . . . . 15 ⊢ ⦋𝐴 / 𝑥⦌(𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (⦋𝐴 / 𝑥⦌𝑦⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)))
28		csbres 5985	. . . . . . . . . . . . . . . . 17 ⊢ ⦋𝐴 / 𝑥⦌(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (⦋𝐴 / 𝑥⦌𝑓 ↾ ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦))
29		csbconstg 3870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑓 = 𝑓)
30	29, 17	reseq12d 5983	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑓 ↾ ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))
31	28, 30	eqtrid 2810	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦)) = (𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))
32	14, 31	oveq12d 7464	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑦⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))
33	27, 32	eqtrid 2810	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))
34	26, 33	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌(𝑓‘𝑦) = ⦋𝐴 / 𝑥⦌(𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ (𝑓‘𝑦) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))))
35	25, 34	bitrd 282	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ (𝑓‘𝑦) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))))
36	35	ralbidv 3189	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ 𝑧 [𝐴 / 𝑥](𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))))
37	24, 36	bitrd 282	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))) ↔ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦)))))
38	5, 23, 37	3anbi123d 1465	. . . . . . . . 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) = (𝑦𝐹(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) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))))
39	4, 38	bitrid 286	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(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) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))))
40	39	exbidv 1956	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (∃𝑧[𝐴 / 𝑥](𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(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) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))))
41	3, 40	bitrid 286	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(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) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))))
42	41	abbidv 2829	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ [𝐴 / 𝑥]∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(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) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))})
43	2, 42	eqtrid 2810	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(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) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))})
44	43	unieqd 4883	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ ⦋𝐴 / 𝑥⦌{𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(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) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))})
45	1, 44	eqtrid 2810	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(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) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))})
46		df-frecs 8330	. . 3 ⊢ frecs(𝑅, 𝐷, 𝐹) = ∪ {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
47	46	csbeq2i 3859	. 2 ⊢ ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, 𝐹) = ⦋𝐴 / 𝑥⦌∪ {𝑓 ∣ ∃𝑧(𝑓 Fn 𝑧 ∧ (𝑧 ⊆ 𝐷 ∧ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐷, 𝑦) ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝑧 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐷, 𝑦))))}
48		df-frecs 8330	. 2 ⊢ frecs(⦋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) = (𝑦⦋𝐴 / 𝑥⦌𝐹(𝑓 ↾ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, 𝑦))))}
49	45, 47, 48	3eqtr4g 2823	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, 𝐹) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  [wsbc 3742  ⦋csb 3851   ⊆ wss 3900  ∪ cuni 4870   ↾ cres 5658  Predcpred 6324   Fn wfn 6556  ‘cfv 6561  (class class class)co 7446  frecscfrecs 8329

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-xp 5662  df-cnv 5664  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-iota 6515  df-fv 6569  df-ov 7449  df-frecs 8330

This theorem is used by:  csbwrecsg  8367