Theorem dfrecs2 34179

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 8174	. 2 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2		elin 3899	. . . . . . . . 9 ⊢ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)))
3		vex 3426	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
4	3	elfuns 34144	. . . . . . . . . 10 ⊢ (𝑓 ∈ Funs ↔ Fun 𝑓)
5		vex 3426	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
6	5, 3	brcnv 5780	. . . . . . . . . . . . 13 ⊢ (𝑥◡Domain𝑓 ↔ 𝑓Domain𝑥)
7	3, 5	brdomain 34162	. . . . . . . . . . . . 13 ⊢ (𝑓Domain𝑥 ↔ 𝑥 = dom 𝑓)
8	6, 7	bitri 274	. . . . . . . . . . . 12 ⊢ (𝑥◡Domain𝑓 ↔ 𝑥 = dom 𝑓)
9	8	rexbii 3177	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ On 𝑥◡Domain𝑓 ↔ ∃𝑥 ∈ On 𝑥 = dom 𝑓)
10	3	elima 5963	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (◡Domain “ On) ↔ ∃𝑥 ∈ On 𝑥◡Domain𝑓)
11		risset 3193	. . . . . . . . . . 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 626	. . . . . . . . 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 5798	. . . . . . . . . . 11 ⊢ (𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∃𝑦 𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦)
16		brdif 5123	. . . . . . . . . . . . 13 ⊢ (𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ (𝑓(◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦))
17		vex 3426	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
18	3, 17	brco 5768	. . . . . . . . . . . . . . 15 ⊢ (𝑓(◡ E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦))
19	7	anbi1i 623	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ (𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
20	19	exbii 1851	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
21	3	dmex 7732	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑓 ∈ V
22		breq1 5073	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = dom 𝑓 → (𝑥◡ E 𝑦 ↔ dom 𝑓◡ E 𝑦))
23	21, 22	ceqsexv 3469	. . . . . . . . . . . . . . . 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 5780	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑓◡ E 𝑦 ↔ 𝑦 E dom 𝑓)
26	21	epeli 5488	. . . . . . . . . . . . . . . 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 5071	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))
30		opex 5373	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
31	30	elfix 34132	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)) ↔ ⟨𝑓, 𝑦⟩(◡Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩)
32	30, 30	brco 5768	. . . . . . . . . . . . . . . . 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 5780	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
35	3, 17, 5	brapply 34167	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑓, 𝑦⟩Apply𝑥 ↔ 𝑥 = (𝑓‘𝑦))
36	34, 35	bitri 274	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓‘𝑦))
37	36	anbi1i 623	. . . . . . . . . . . . . . . . . . . 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 1851	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ ∃𝑥(𝑥 = (𝑓‘𝑦) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥))
40		fvex 6769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑦) ∈ V
41		breq2 5074	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑓‘𝑦) → (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦)))
42	40, 41	ceqsexv 3469	. . . . . . . . . . . . . . . . . 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 5768	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦) ↔ ∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)))
45	3, 17, 5	brrestrict 34178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑓, 𝑦⟩Restrict𝑥 ↔ 𝑥 = (𝑓 ↾ 𝑦))
46	45	anbi1i 623	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ (𝑥 = (𝑓 ↾ 𝑦) ∧ 𝑥FullFun𝐹(𝑓‘𝑦)))
47	46	exbii 1851	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)) ↔ ∃𝑥(𝑥 = (𝑓 ↾ 𝑦) ∧ 𝑥FullFun𝐹(𝑓‘𝑦)))
48	3	resex 5928	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ↾ 𝑦) ∈ V
49		breq1 5073	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑓 ↾ 𝑦) → (𝑥FullFun𝐹(𝑓‘𝑦) ↔ (𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦)))
50	48, 49	ceqsexv 3469	. . . . . . . . . . . . . . . . . . 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 34177	. . . . . . . . . . . . . . . . . 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 626	. . . . . . . . . . . . 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 1851	. . . . . . . . . . 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 3165	. . . . . . . . . 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 356	. . . . . . . 8 ⊢ (¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
65	14, 64	anbi12i 626	. . . . . . 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 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 2826	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
69		raleq 3333	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
70	68, 69	anbi12d 630	. . . . . . . 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 628	. . . . . . 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 3469	. . . . . 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 6421	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝑥))
74		eqcom 2745	. . . . . . . . . . 11 ⊢ (dom 𝑓 = 𝑥 ↔ 𝑥 = dom 𝑓)
75	74	anbi2i 622	. . . . . . . . . 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 296	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑥 ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
78	77	anbi1i 623	. . . . . . . 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 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 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 1851	. . . . . 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 3893	. . . . 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	abbi2i 2878	. . 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 4849	. 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 2769	1 ⊢ recs(𝐹) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ∩ cin 3882  ⟨cop 4564  ∪ cuni 4836   class class class wbr 5070   E cep 5485  ^◡ccnv 5579  dom cdm 5580   ↾ cres 5582   “ cima 5583   ∘ ccom 5584  Oncon0 6251  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418  recscrecs 8172   Fix cfix 34064   Funs cfuns 34066  Domaincdomain 34072  Applycapply 34074  FullFuncfullfn 34079  Restrictcrestrict 34080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-symdif 4173  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-ov 7258  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-txp 34083  df-pprod 34084  df-bigcup 34087  df-fix 34088  df-funs 34090  df-singleton 34091  df-singles 34092  df-image 34093  df-cart 34094  df-img 34095  df-domain 34096  df-range 34097  df-cap 34099  df-restrict 34100  df-apply 34102  df-funpart 34103  df-fullfun 34104

This theorem is referenced by: (None)