Theorem dfrecs2 36714

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 8380	. 2 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2		elin 3915	. . . . . . . . 9 ⊢ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)))
3		vex 3455	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
4	3	elfuns 36677	. . . . . . . . . 10 ⊢ (𝑓 ∈ Funs ↔ Fun 𝑓)
5		vex 3455	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
6	5, 3	brcnv 5860	. . . . . . . . . . . . 13 ⊢ (𝑥◡Domain𝑓 ↔ 𝑓Domain𝑥)
7	3, 5	brdomain 36695	. . . . . . . . . . . . 13 ⊢ (𝑓Domain𝑥 ↔ 𝑥 = dom 𝑓)
8	6, 7	bitri 278	. . . . . . . . . . . 12 ⊢ (𝑥◡Domain𝑓 ↔ 𝑥 = dom 𝑓)
9	8	rexbii 3110	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ On 𝑥◡Domain𝑓 ↔ ∃𝑥 ∈ On 𝑥 = dom 𝑓)
10	3	elima 6061	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (◡Domain “ On) ↔ ∃𝑥 ∈ On 𝑥◡Domain𝑓)
11		risset 3238	. . . . . . . . . . 11 ⊢ (dom 𝑓 ∈ On ↔ ∃𝑥 ∈ On 𝑥 = dom 𝑓)
12	9, 10, 11	3bitr4i 306	. . . . . . . . . 10 ⊢ (𝑓 ∈ (◡Domain “ On) ↔ dom 𝑓 ∈ On)
13	4, 12	anbi12i 640	. . . . . . . . 9 ⊢ ((𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
14	2, 13	bitri 278	. . . . . . . 8 ⊢ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
15	3	eldm 5882	. . . . . . . . . . 11 ⊢ (𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∃𝑦 𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦)
16		brdif 5158	. . . . . . . . . . . . 13 ⊢ (𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ (𝑓(◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦))
17		vex 3455	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
18	3, 17	brco 5848	. . . . . . . . . . . . . . 15 ⊢ (𝑓(◡ E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦))
19	7	anbi1i 636	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ (𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
20	19	exbii 1881	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
21	3	dmex 7921	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑓 ∈ V
22		breq1 5106	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = dom 𝑓 → (𝑥◡ E 𝑦 ↔ dom 𝑓◡ E 𝑦))
23	21, 22	ceqsexv 3499	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦) ↔ dom 𝑓◡ E 𝑦)
24	20, 23	bitri 278	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ dom 𝑓◡ E 𝑦)
25	21, 17	brcnv 5860	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑓◡ E 𝑦 ↔ 𝑦 E dom 𝑓)
26	21	epeli 5553	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 E dom 𝑓 ↔ 𝑦 ∈ dom 𝑓)
27	25, 26	bitri 278	. . . . . . . . . . . . . . 15 ⊢ (dom 𝑓◡ E 𝑦 ↔ 𝑦 ∈ dom 𝑓)
28	18, 24, 27	3bitri 300	. . . . . . . . . . . . . 14 ⊢ (𝑓(◡ E ∘ Domain)𝑦 ↔ 𝑦 ∈ dom 𝑓)
29		df-br 5104	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))
30		opex 5432	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
31	30	elfix 36665	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)) ↔ ⟨𝑓, 𝑦⟩(◡Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩)
32	30, 30	brco 5848	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑓, 𝑦⟩(◡Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩ ↔ ∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩))
33		ancom 466	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ (𝑥◡Apply⟨𝑓, 𝑦⟩ ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
34	5, 30	brcnv 5860	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
35	3, 17, 5	brapply 36700	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑓, 𝑦⟩Apply𝑥 ↔ 𝑥 = (𝑓‘𝑦))
36	34, 35	bitri 278	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓‘𝑦))
37	36	anbi1i 636	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥◡Apply⟨𝑓, 𝑦⟩ ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥) ↔ (𝑥 = (𝑓‘𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
38	33, 37	bitri 278	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ (𝑥 = (𝑓‘𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
39	38	exbii 1881	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ ∃𝑥(𝑥 = (𝑓‘𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
40		fvex 6898	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑦) ∈ V
41		breq2 5107	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑓‘𝑦) → (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦)))
42	40, 41	ceqsexv 3499	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(𝑥 = (𝑓‘𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥) ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦))
43	39, 42	bitri 278	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦))
44	30, 40	brco 5848	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦) ↔ ∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)))
45	3, 17, 5	brrestrict 36713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑓, 𝑦⟩Restrict𝑥 ↔ 𝑥 = (𝑓 ↾ 𝑦))
46	45	anbi1i 636	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ (𝑥 = (𝑓 ↾ 𝑦) ∧ 𝑥FullFun𝐹(𝑓‘𝑦)))
47	46	exbii 1881	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ ∃𝑥(𝑥 = (𝑓 ↾ 𝑦) ∧ 𝑥FullFun𝐹(𝑓‘𝑦)))
48	3	resex 6018	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ↾ 𝑦) ∈ V
49		breq1 5106	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑓 ↾ 𝑦) → (𝑥FullFun𝐹(𝑓‘𝑦) ↔ (𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦)))
50	48, 49	ceqsexv 3499	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥(𝑥 = (𝑓 ↾ 𝑦) ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ (𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦))
51	47, 50	bitri 278	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ (𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦))
52	48, 40	brfullfun 36712	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
53	44, 51, 52	3bitri 300	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
54	32, 43, 53	3bitri 300	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑓, 𝑦⟩(◡Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩ ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
55	29, 31, 54	3bitri 300	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
56	55	notbii 323	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
57	28, 56	anbi12i 640	. . . . . . . . . . . . 13 ⊢ ((𝑓(◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦) ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
58	16, 57	bitri 278	. . . . . . . . . . . 12 ⊢ (𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
59	58	exbii 1881	. . . . . . . . . . 11 ⊢ (∃𝑦 𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
60	15, 59	bitri 278	. . . . . . . . . 10 ⊢ (𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
61		df-rex 3088	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ dom 𝑓 ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
62		rexnal 3115	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ dom 𝑓 ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ¬ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
63	60, 61, 62	3bitr2ri 303	. . . . . . . . 9 ⊢ (¬ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))))
64	63	con1bii 359	. . . . . . . 8 ⊢ (¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
65	14, 64	anbi12i 640	. . . . . . 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 474	. . . . . . 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 278	. . . . . 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 2849	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
69		raleq 3317	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
70	68, 69	anbi12d 644	. . . . . . . 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 642	. . . . . . 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 3499	. . . . . 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 6541	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝑥))
74		eqcom 2768	. . . . . . . . . . 11 ⊢ (dom 𝑓 = 𝑥 ↔ 𝑥 = dom 𝑓)
75	74	anbi2i 635	. . . . . . . . . 10 ⊢ ((Fun 𝑓 ∧ dom 𝑓 = 𝑥) ↔ (Fun 𝑓 ∧ 𝑥 = dom 𝑓))
76		ancom 466	. . . . . . . . . 10 ⊢ ((Fun 𝑓 ∧ 𝑥 = dom 𝑓) ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
77	73, 75, 76	3bitri 300	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑥 ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
78	77	anbi1i 636	. . . . . . . 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 658	. . . . . . . 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 474	. . . . . . . 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 305	. . . . . . 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 1881	. . . . . 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 302	. . . . 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 3909	. . . . 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 3088	. . . . 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 306	. . . 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 2896	. . 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 4879	. 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 2787	1 ⊢ recs(𝐹) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087   ∖ cdif 3896   ∩ cin 3898  ⟨cop 4590  ∪ cuni 4867   class class class wbr 5103   E cep 5550  ^◡ccnv 5650  dom cdm 5651   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  Oncon0 6362  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  recscrecs 8378   Fix cfix 36597   Funs cfuns 36599  Domaincdomain 36605  Applycapply 36607  FullFuncfullfn 36612  Restrictcrestrict 36613

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-symdif 4199  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-ov 7423  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-txp 36616  df-pprod 36617  df-bigcup 36620  df-fix 36621  df-funs 36623  df-singleton 36624  df-singles 36625  df-image 36626  df-cart 36627  df-img 36628  df-domain 36629  df-range 36630  df-cap 36632  df-restrict 36633  df-apply 36635  df-funpart 36636  df-fullfun 36637

This theorem is used by: (None)