Theorem dfrecs2 35938

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 8341	. 2 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2		elin 3930	. . . . . . . . 9 ⊢ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)))
3		vex 3451	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
4	3	elfuns 35903	. . . . . . . . . 10 ⊢ (𝑓 ∈ Funs ↔ Fun 𝑓)
5		vex 3451	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
6	5, 3	brcnv 5846	. . . . . . . . . . . . 13 ⊢ (𝑥◡Domain𝑓 ↔ 𝑓Domain𝑥)
7	3, 5	brdomain 35921	. . . . . . . . . . . . 13 ⊢ (𝑓Domain𝑥 ↔ 𝑥 = dom 𝑓)
8	6, 7	bitri 275	. . . . . . . . . . . 12 ⊢ (𝑥◡Domain𝑓 ↔ 𝑥 = dom 𝑓)
9	8	rexbii 3076	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ On 𝑥◡Domain𝑓 ↔ ∃𝑥 ∈ On 𝑥 = dom 𝑓)
10	3	elima 6036	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (◡Domain “ On) ↔ ∃𝑥 ∈ On 𝑥◡Domain𝑓)
11		risset 3212	. . . . . . . . . . 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 5864	. . . . . . . . . . 11 ⊢ (𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))) ↔ ∃𝑦 𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦)
16		brdif 5160	. . . . . . . . . . . . 13 ⊢ (𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))𝑦 ↔ (𝑓(◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦))
17		vex 3451	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
18	3, 17	brco 5834	. . . . . . . . . . . . . . 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 7885	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑓 ∈ V
22		breq1 5110	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = dom 𝑓 → (𝑥◡ E 𝑦 ↔ dom 𝑓◡ E 𝑦))
23	21, 22	ceqsexv 3498	. . . . . . . . . . . . . . . 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 5846	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑓◡ E 𝑦 ↔ 𝑦 E dom 𝑓)
26	21	epeli 5540	. . . . . . . . . . . . . . . 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 5108	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))
30		opex 5424	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
31	30	elfix 35891	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)) ↔ ⟨𝑓, 𝑦⟩(◡Apply ∘ (FullFun𝐹 ∘ Restrict))⟨𝑓, 𝑦⟩)
32	30, 30	brco 5834	. . . . . . . . . . . . . . . . 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 5846	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
35	3, 17, 5	brapply 35926	. . . . . . . . . . . . . . . . . . . . . 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 6871	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑦) ∈ V
41		breq2 5111	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑓‘𝑦) → (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)𝑥 ↔ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦)))
42	40, 41	ceqsexv 3498	. . . . . . . . . . . . . . . . . 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 5834	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ Restrict)(𝑓‘𝑦) ↔ ∃𝑥(⟨𝑓, 𝑦⟩Restrict𝑥 ∧ 𝑥FullFun𝐹(𝑓‘𝑦)))
45	3, 17, 5	brrestrict 35937	. . . . . . . . . . . . . . . . . . . . 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 6000	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ↾ 𝑦) ∈ V
49		breq1 5110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑓 ↾ 𝑦) → (𝑥FullFun𝐹(𝑓‘𝑦) ↔ (𝑓 ↾ 𝑦)FullFun𝐹(𝑓‘𝑦)))
50	48, 49	ceqsexv 3498	. . . . . . . . . . . . . . . . . . 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 35936	. . . . . . . . . . . . . . . . . 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 3054	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ dom 𝑓 ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∃𝑦(𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
62		rexnal 3082	. . . . . . . . . 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 2816	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
69		raleq 3296	. . . . . . . . 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 3498	. . . . . 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 6514	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝑥))
74		eqcom 2736	. . . . . . . . . . 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 3924	. . . . 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 3054	. . . . 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 2863	. . 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 4883	. 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 2755	1 ⊢ recs(𝐹) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict))))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053   ∖ cdif 3911   ∩ cin 3913  ⟨cop 4595  ∪ cuni 4871   class class class wbr 5107   E cep 5537  ^◡ccnv 5637  dom cdm 5638   ↾ cres 5640   “ cima 5641   ∘ ccom 5642  Oncon0 6332  Fun wfun 6505   Fn wfn 6506  ‘cfv 6511  recscrecs 8339   Fix cfix 35823   Funs cfuns 35825  Domaincdomain 35831  Applycapply 35833  FullFuncfullfn 35838  Restrictcrestrict 35839

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-symdif 4216  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-ov 7390  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-txp 35842  df-pprod 35843  df-bigcup 35846  df-fix 35847  df-funs 35849  df-singleton 35850  df-singles 35851  df-image 35852  df-cart 35853  df-img 35854  df-domain 35855  df-range 35856  df-cap 35858  df-restrict 35859  df-apply 35861  df-funpart 35862  df-fullfun 35863

This theorem is referenced by: (None)