dfrecs2 - Mathbox for Scott Fenton

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Theorem dfrecs2 35226

Description: A quantifier-free definition of recs. (Contributed by Scott Fenton, 17-Jul-2020.)

**Assertion**
Ref	Expression
dfrecs2	⊢ recs(𝐹) = ∪ (( Funs ∩ (^◡Domain “ On)) ∖ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict))))

Proof of Theorem dfrecs2 Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrecs3 8374	. 2 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2		elin 3963	. . . . . . . . 9 ⊢ (𝑓 ∈ ( Funs ∩ (^◡Domain “ On)) ↔ (𝑓 ∈ Funs ∧ 𝑓 ∈ (^◡Domain “ On)))
3		vex 3476	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
4	3	elfuns 35191	. . . . . . . . . 10 ⊢ (𝑓 ∈ Funs ↔ Fun 𝑓)
5		vex 3476	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
6	5, 3	brcnv 5881	. . . . . . . . . . . . 13 ⊢ (𝑥^◡Domain𝑓 ↔ 𝑓Domain𝑥)
7	3, 5	brdomain 35209	. . . . . . . . . . . . 13 ⊢ (𝑓Domain𝑥 ↔ 𝑥 = dom 𝑓)
8	6, 7	bitri 274	. . . . . . . . . . . 12 ⊢ (𝑥^◡Domain𝑓 ↔ 𝑥 = dom 𝑓)
9	8	rexbii 3092	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ On 𝑥^◡Domain𝑓 ↔ ∃𝑥 ∈ On 𝑥 = dom 𝑓)
10	3	elima 6063	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (^◡Domain “ On) ↔ ∃𝑥 ∈ On 𝑥^◡Domain𝑓)
11		risset 3228	. . . . . . . . . . 11 ⊢ (dom 𝑓 ∈ On ↔ ∃𝑥 ∈ On 𝑥 = dom 𝑓)
12	9, 10, 11	3bitr4i 302	. . . . . . . . . 10 ⊢ (𝑓 ∈ (^◡Domain “ On) ↔ dom 𝑓 ∈ On)
13	4, 12	anbi12i 625	. . . . . . . . 9 ⊢ ((𝑓 ∈ Funs ∧ 𝑓 ∈ (^◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
14	2, 13	bitri 274	. . . . . . . 8 ⊢ (𝑓 ∈ ( Funs ∩ (^◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
15	3	eldm 5899	. . . . . . . . . . 11 ⊢ (𝑓 ∈ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∃𝑦 𝑓((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦)
16		brdif 5200	. . . . . . . . . . . . 13 ⊢ (𝑓((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ (𝑓(^◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦))
17		vex 3476	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
18	3, 17	brco 5869	. . . . . . . . . . . . . . 15 ⊢ (𝑓(^◡ E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥 ∧ 𝑥^◡ E 𝑦))
19	7	anbi1i 622	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓Domain𝑥 ∧ 𝑥^◡ E 𝑦) ↔ (𝑥 = dom 𝑓 ∧ 𝑥^◡ E 𝑦))
20	19	exbii 1848	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥(𝑓Domain𝑥 ∧ 𝑥^◡ E 𝑦) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥^◡ E 𝑦))
21	3	dmex 7904	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑓 ∈ V
22		breq1 5150	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = dom 𝑓 → (𝑥^◡ E 𝑦 ↔ dom 𝑓^◡ E 𝑦))
23	21, 22	ceqsexv 3524	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥^◡ E 𝑦) ↔ dom 𝑓^◡ E 𝑦)
24	20, 23	bitri 274	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥(𝑓Domain𝑥 ∧ 𝑥^◡ E 𝑦) ↔ dom 𝑓^◡ E 𝑦)
25	21, 17	brcnv 5881	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑓^◡ E 𝑦 ↔ 𝑦 E dom 𝑓)
26	21	epeli 5581	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 E dom 𝑓 ↔ 𝑦 ∈ dom 𝑓)
27	25, 26	bitri 274	. . . . . . . . . . . . . . 15 ⊢ (dom 𝑓^◡ E 𝑦 ↔ 𝑦 ∈ dom 𝑓)
28	18, 24, 27	3bitri 296	. . . . . . . . . . . . . 14 ⊢ (𝑓(^◡ E ∘ Domain)𝑦 ↔ 𝑦 ∈ dom 𝑓)
29		df-br 5148	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))
30		opex 5463	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
31	30	elfix 35179	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑓, 𝑦⟩ ∈ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)) ↔ ⟨𝑓, 𝑦⟩(^◡Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩)
32	30, 30	brco 5869	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑓, 𝑦⟩(^◡Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩ ↔ ∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥^◡Apply⟨𝑓, 𝑦⟩))
33		ancom 459	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥^◡Apply⟨𝑓, 𝑦⟩) ↔ (𝑥^◡Apply⟨𝑓, 𝑦⟩ ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
34	5, 30	brcnv 5881	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥^◡Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
35	3, 17, 5	brapply 35214	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑓, 𝑦⟩Apply𝑥 ↔ 𝑥 = (𝑓‘𝑦))
36	34, 35	bitri 274	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥^◡Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓‘𝑦))
37	36	anbi1i 622	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥^◡Apply⟨𝑓, 𝑦⟩ ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥) ↔ (𝑥 = (𝑓‘𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
38	33, 37	bitri 274	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥^◡Apply⟨𝑓, 𝑦⟩) ↔ (𝑥 = (𝑓‘𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
39	38	exbii 1848	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥^◡Apply⟨𝑓, 𝑦⟩) ↔ ∃𝑥(𝑥 = (𝑓‘𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
40		fvex 6903	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑦) ∈ V
41		breq2 5151	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑓‘𝑦) → (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦)))
42	40, 41	ceqsexv 3524	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(𝑥 = (𝑓‘𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥) ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦))
43	39, 42	bitri 274	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥^◡Apply⟨𝑓, 𝑦⟩) ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦))
44	30, 40	brco 5869	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦) ↔ ∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)))
45	3, 17, 5	brrestrict 35225	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑓, 𝑦⟩Restrict𝑥 ↔ 𝑥 = (𝑓 ↾ 𝑦))
46	45	anbi1i 622	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ (𝑥 = (𝑓 ↾ 𝑦) ∧ 𝑥FullFun𝐹(𝑓‘𝑦)))
47	46	exbii 1848	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ ∃𝑥(𝑥 = (𝑓 ↾ 𝑦) ∧ 𝑥FullFun𝐹(𝑓‘𝑦)))
48	3	resex 6028	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ↾ 𝑦) ∈ V
49		breq1 5150	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑓 ↾ 𝑦) → (𝑥FullFun𝐹(𝑓‘𝑦) ↔ (𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦)))
50	48, 49	ceqsexv 3524	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥(𝑥 = (𝑓 ↾ 𝑦) ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ (𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦))
51	47, 50	bitri 274	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ (𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦))
52	48, 40	brfullfun 35224	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
53	44, 51, 52	3bitri 296	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
54	32, 43, 53	3bitri 296	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑓, 𝑦⟩(^◡Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩ ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
55	29, 31, 54	3bitri 296	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
56	55	notbii 319	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑓 Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
57	28, 56	anbi12i 625	. . . . . . . . . . . . 13 ⊢ ((𝑓(^◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦) ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
58	16, 57	bitri 274	. . . . . . . . . . . 12 ⊢ (𝑓((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
59	58	exbii 1848	. . . . . . . . . . 11 ⊢ (∃𝑦 𝑓((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
60	15, 59	bitri 274	. . . . . . . . . 10 ⊢ (𝑓 ∈ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
61		df-rex 3069	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ dom 𝑓 ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
62		rexnal 3098	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ dom 𝑓 ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ¬ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
63	60, 61, 62	3bitr2ri 299	. . . . . . . . 9 ⊢ (¬ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ 𝑓 ∈ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict))))
64	63	con1bii 355	. . . . . . . 8 ⊢ (¬ 𝑓 ∈ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
65	14, 64	anbi12i 625	. . . . . . 7 ⊢ ((𝑓 ∈ ( Funs ∩ (^◡Domain “ On)) ∧ ¬ 𝑓 ∈ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
66		anass 467	. . . . . . 7 ⊢ (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
67	65, 66	bitri 274	. . . . . 6 ⊢ ((𝑓 ∈ ( Funs ∩ (^◡Domain “ On)) ∧ ¬ 𝑓 ∈ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
68		eleq1 2819	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
69		raleq 3320	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
70	68, 69	anbi12d 629	. . . . . . . 8 ⊢ (𝑥 = dom 𝑓 → ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) ↔ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
71	70	anbi2d 627	. . . . . . 7 ⊢ (𝑥 = dom 𝑓 → ((Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))))
72	21, 71	ceqsexv 3524	. . . . . 6 ⊢ (∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
73		df-fn 6545	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝑥))
74		eqcom 2737	. . . . . . . . . . 11 ⊢ (dom 𝑓 = 𝑥 ↔ 𝑥 = dom 𝑓)
75	74	anbi2i 621	. . . . . . . . . 10 ⊢ ((Fun 𝑓 ∧ dom 𝑓 = 𝑥) ↔ (Fun 𝑓 ∧ 𝑥 = dom 𝑓))
76		ancom 459	. . . . . . . . . 10 ⊢ ((Fun 𝑓 ∧ 𝑥 = dom 𝑓) ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
77	73, 75, 76	3bitri 296	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑥 ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
78	77	anbi1i 622	. . . . . . . 8 ⊢ ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) ↔ ((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
79		an12 641	. . . . . . . 8 ⊢ ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
80		anass 467	. . . . . . . 8 ⊢ (((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) ↔ (𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))))
81	78, 79, 80	3bitr3ri 301	. . . . . . 7 ⊢ ((𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
82	81	exbii 1848	. . . . . 6 ⊢ (∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
83	67, 72, 82	3bitr2i 298	. . . . 5 ⊢ ((𝑓 ∈ ( Funs ∩ (^◡Domain “ On)) ∧ ¬ 𝑓 ∈ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
84		eldif 3957	. . . . 5 ⊢ (𝑓 ∈ (( Funs ∩ (^◡Domain “ On)) ∖ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ (𝑓 ∈ ( Funs ∩ (^◡Domain “ On)) ∧ ¬ 𝑓 ∈ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))))
85		df-rex 3069	. . . . 5 ⊢ (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
86	83, 84, 85	3bitr4i 302	. . . 4 ⊢ (𝑓 ∈ (( Funs ∩ (^◡Domain “ On)) ∖ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
87	86	eqabi 2867	. . 3 ⊢ (( Funs ∩ (^◡Domain “ On)) ∖ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
88	87	unieqi 4920	. 2 ⊢ ∪ (( Funs ∩ (^◡Domain “ On)) ∖ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict)))) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
89	1, 88	eqtr4i 2761	1 ⊢ recs(𝐹) = ∪ (( Funs ∩ (^◡Domain “ On)) ∖ dom ((^◡ E ∘ Domain) ∖ Fix (^◡Apply ∘ (FullFun𝐹 ∘ Restrict))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 394 = wceq 1539 ∃wex 1779 ∈ wcel 2104 {cab 2707 ∀wral 3059 ∃wrex 3068 ∖ cdif 3944 ∩ cin 3946 ⟨cop 4633 ∪ cuni 4907 class class class wbr 5147 E cep 5578 ^◡ccnv 5674 dom cdm 5675 ↾ cres 5677 “ cima 5678 ∘ ccom 5679 Oncon0 6363 Fun wfun 6536 Fn wfn 6537 ‘cfv 6542 recscrecs 8372 Fix cfix 35111 Funs cfuns 35113 Domaincdomain 35119 Applycapply 35121 FullFuncfullfn 35126 Restrictcrestrict 35127

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-symdif 4241 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fo 6548 df-fv 6550 df-ov 7414 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-txp 35130 df-pprod 35131 df-bigcup 35134 df-fix 35135 df-funs 35137 df-singleton 35138 df-singles 35139 df-image 35140 df-cart 35141 df-img 35142 df-domain 35143 df-range 35144 df-cap 35146 df-restrict 35147 df-apply 35149 df-funpart 35150 df-fullfun 35151

This theorem is referenced by: (None)

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