Theorem dfrecs2 35951

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 8412	. 2 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2		elin 3967	. . . . . . . . 9 ⊢ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)))
3		vex 3484	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
4	3	elfuns 35916	. . . . . . . . . 10 ⊢ (𝑓 ∈ Funs ↔ Fun 𝑓)
5		vex 3484	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
6	5, 3	brcnv 5893	. . . . . . . . . . . . 13 ⊢ (𝑥◡Domain𝑓 ↔ 𝑓Domain𝑥)
7	3, 5	brdomain 35934	. . . . . . . . . . . . 13 ⊢ (𝑓Domain𝑥 ↔ 𝑥 = dom 𝑓)
8	6, 7	bitri 275	. . . . . . . . . . . 12 ⊢ (𝑥◡Domain𝑓 ↔ 𝑥 = dom 𝑓)
9	8	rexbii 3094	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ On 𝑥◡Domain𝑓 ↔ ∃𝑥 ∈ On 𝑥 = dom 𝑓)
10	3	elima 6083	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (◡Domain “ On) ↔ ∃𝑥 ∈ On 𝑥◡Domain𝑓)
11		risset 3233	. . . . . . . . . . 11 ⊢ (dom 𝑓 ∈ On ↔ ∃𝑥 ∈ On 𝑥 = dom 𝑓)
12	9, 10, 11	3bitr4i 303	. . . . . . . . . 10 ⊢ (𝑓 ∈ (◡Domain “ On) ↔ dom 𝑓 ∈ On)
13	4, 12	anbi12i 628	. . . . . . . . 9 ⊢ ((𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
14	2, 13	bitri 275	. . . . . . . 8 ⊢ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
15	3	eldm 5911	. . . . . . . . . . 11 ⊢ (𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∃𝑦 𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦)
16		brdif 5196	. . . . . . . . . . . . 13 ⊢ (𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ (𝑓(◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦))
17		vex 3484	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
18	3, 17	brco 5881	. . . . . . . . . . . . . . 15 ⊢ (𝑓(◡ E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦))
19	7	anbi1i 624	. . . . . . . . . . . . . . . . 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 7931	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑓 ∈ V
22		breq1 5146	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = dom 𝑓 → (𝑥◡ E 𝑦 ↔ dom 𝑓◡ E 𝑦))
23	21, 22	ceqsexv 3532	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦) ↔ dom 𝑓◡ E 𝑦)
24	20, 23	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ dom 𝑓◡ E 𝑦)
25	21, 17	brcnv 5893	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑓◡ E 𝑦 ↔ 𝑦 E dom 𝑓)
26	21	epeli 5586	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 E dom 𝑓 ↔ 𝑦 ∈ dom 𝑓)
27	25, 26	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (dom 𝑓◡ E 𝑦 ↔ 𝑦 ∈ dom 𝑓)
28	18, 24, 27	3bitri 297	. . . . . . . . . . . . . 14 ⊢ (𝑓(◡ E ∘ Domain)𝑦 ↔ 𝑦 ∈ dom 𝑓)
29		df-br 5144	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))
30		opex 5469	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
31	30	elfix 35904	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)) ↔ ⟨𝑓, 𝑦⟩(◡Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩)
32	30, 30	brco 5881	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑓, 𝑦⟩(◡Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩ ↔ ∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩))
33		ancom 460	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ (𝑥◡Apply⟨𝑓, 𝑦⟩ ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
34	5, 30	brcnv 5893	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
35	3, 17, 5	brapply 35939	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑓, 𝑦⟩Apply𝑥 ↔ 𝑥 = (𝑓‘𝑦))
36	34, 35	bitri 275	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓‘𝑦))
37	36	anbi1i 624	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥◡Apply⟨𝑓, 𝑦⟩ ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥) ↔ (𝑥 = (𝑓‘𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
38	33, 37	bitri 275	. . . . . . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑦) ∈ V
41		breq2 5147	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑓‘𝑦) → (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦)))
42	40, 41	ceqsexv 3532	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(𝑥 = (𝑓‘𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥) ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦))
43	39, 42	bitri 275	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦))
44	30, 40	brco 5881	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦) ↔ ∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)))
45	3, 17, 5	brrestrict 35950	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑓, 𝑦⟩Restrict𝑥 ↔ 𝑥 = (𝑓 ↾ 𝑦))
46	45	anbi1i 624	. . . . . . . . . . . . . . . . . . . 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 6047	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ↾ 𝑦) ∈ V
49		breq1 5146	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑓 ↾ 𝑦) → (𝑥FullFun𝐹(𝑓‘𝑦) ↔ (𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦)))
50	48, 49	ceqsexv 3532	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥(𝑥 = (𝑓 ↾ 𝑦) ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ (𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦))
51	47, 50	bitri 275	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ (𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦))
52	48, 40	brfullfun 35949	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
53	44, 51, 52	3bitri 297	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
54	32, 43, 53	3bitri 297	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑓, 𝑦⟩(◡Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩ ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
55	29, 31, 54	3bitri 297	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
56	55	notbii 320	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
57	28, 56	anbi12i 628	. . . . . . . . . . . . 13 ⊢ ((𝑓(◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦) ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
58	16, 57	bitri 275	. . . . . . . . . . . 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 275	. . . . . . . . . 10 ⊢ (𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
61		df-rex 3071	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ dom 𝑓 ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
62		rexnal 3100	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ dom 𝑓 ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ¬ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
63	60, 61, 62	3bitr2ri 300	. . . . . . . . 9 ⊢ (¬ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))))
64	63	con1bii 356	. . . . . . . 8 ⊢ (¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
65	14, 64	anbi12i 628	. . . . . . 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 468	. . . . . . 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 275	. . . . . 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 2829	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
69		raleq 3323	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
70	68, 69	anbi12d 632	. . . . . . . 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 630	. . . . . . 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 3532	. . . . . 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 6564	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝑥))
74		eqcom 2744	. . . . . . . . . . 11 ⊢ (dom 𝑓 = 𝑥 ↔ 𝑥 = dom 𝑓)
75	74	anbi2i 623	. . . . . . . . . 10 ⊢ ((Fun 𝑓 ∧ dom 𝑓 = 𝑥) ↔ (Fun 𝑓 ∧ 𝑥 = dom 𝑓))
76		ancom 460	. . . . . . . . . 10 ⊢ ((Fun 𝑓 ∧ 𝑥 = dom 𝑓) ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
77	73, 75, 76	3bitri 297	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑥 ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
78	77	anbi1i 624	. . . . . . . 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 645	. . . . . . . 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 468	. . . . . . . 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 302	. . . . . . 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 299	. . . . 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 3961	. . . . 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 3071	. . . . 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 303	. . . 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 2877	. . 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 4919	. 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 2768	1 ⊢ recs(𝐹) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948   ∩ cin 3950  ⟨cop 4632  ∪ cuni 4907   class class class wbr 5143   E cep 5583  ^◡ccnv 5684  dom cdm 5685   ↾ cres 5687   “ cima 5688   ∘ ccom 5689  Oncon0 6384  Fun wfun 6555   Fn wfn 6556  ‘cfv 6561  recscrecs 8410   Fix cfix 35836   Funs cfuns 35838  Domaincdomain 35844  Applycapply 35846  FullFuncfullfn 35851  Restrictcrestrict 35852

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-symdif 4253  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-txp 35855  df-pprod 35856  df-bigcup 35859  df-fix 35860  df-funs 35862  df-singleton 35863  df-singles 35864  df-image 35865  df-cart 35866  df-img 35867  df-domain 35868  df-range 35869  df-cap 35871  df-restrict 35872  df-apply 35874  df-funpart 35875  df-fullfun 35876

This theorem is referenced by: (None)