Theorem fucoppc 49769

Description: The isomorphism from the opposite category of functors to the category of opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)

Hypotheses

Ref	Expression
fucoppc.o	⊢ 𝑂 = (oppCat‘𝐶)
fucoppc.p	⊢ 𝑃 = (oppCat‘𝐷)
fucoppc.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fucoppc.r	⊢ 𝑅 = (oppCat‘𝑄)
fucoppc.s	⊢ 𝑆 = (𝑂 FuncCat 𝑃)
fucoppc.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fucoppc.f	⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
fucoppc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
fucoppc.t	⊢ 𝑇 = (CatCat‘𝑈)
fucoppc.b	⊢ 𝐵 = (Base‘𝑇)
fucoppc.i	⊢ 𝐼 = (Iso‘𝑇)
fucoppc.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
fucoppc.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)
fucoppc.1	⊢ (𝜑 → 𝑅 ∈ 𝐵)
fucoppc.2	⊢ (𝜑 → 𝑆 ∈ 𝐵)

Assertion

Ref	Expression
fucoppc	⊢ (𝜑 → 𝐹(𝑅𝐼𝑆)𝐺)

Distinct variable groups:   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝑁,𝑦   𝑥,𝑅,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝑃(𝑥,𝑦)   𝑄(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑇(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐼(𝑥,𝑦)   𝑂(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem fucoppc

Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucoppc.r	. . . . . . 7 ⊢ 𝑅 = (oppCat‘𝑄)
2		fucoppc.q	. . . . . . . 8 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
3	2	fucbas 17899	. . . . . . 7 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
4	1, 3	oppcbas 17653	. . . . . 6 ⊢ (𝐶 Func 𝐷) = (Base‘𝑅)
5		fucoppc.s	. . . . . . 7 ⊢ 𝑆 = (𝑂 FuncCat 𝑃)
6	5	fucbas 17899	. . . . . 6 ⊢ (𝑂 Func 𝑃) = (Base‘𝑆)
7		eqid 2737	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
8		eqid 2737	. . . . . . 7 ⊢ (𝑂 Nat 𝑃) = (𝑂 Nat 𝑃)
9	5, 8	fuchom 17900	. . . . . 6 ⊢ (𝑂 Nat 𝑃) = (Hom ‘𝑆)
10		eqid 2737	. . . . . 6 ⊢ (Id‘𝑅) = (Id‘𝑅)
11		eqid 2737	. . . . . 6 ⊢ (Id‘𝑆) = (Id‘𝑆)
12		eqid 2737	. . . . . 6 ⊢ (comp‘𝑅) = (comp‘𝑅)
13		eqid 2737	. . . . . 6 ⊢ (comp‘𝑆) = (comp‘𝑆)
14		fucoppc.1	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
15		fucoppc.t	. . . . . . . . 9 ⊢ 𝑇 = (CatCat‘𝑈)
16		fucoppc.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑇)
17	15, 16	elbasfv 17154	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝐵 → 𝑈 ∈ V)
18	14, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
19	15, 16, 18	catcbas 18037	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
20	14, 19	eleqtrd 2839	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝑈 ∩ Cat))
21	20	elin2d 4159	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Cat)
22		fucoppc.2	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
23	22, 19	eleqtrd 2839	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (𝑈 ∩ Cat))
24	23	elin2d 4159	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Cat)
25		fucoppc.o	. . . . . . . . 9 ⊢ 𝑂 = (oppCat‘𝐶)
26		fucoppc.p	. . . . . . . . 9 ⊢ 𝑃 = (oppCat‘𝐷)
27		fucoppc.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
28		fucoppc.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
29	25, 26, 27, 28	oppff1o 49508	. . . . . . . 8 ⊢ (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))
30		fucoppc.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
31	30	f1oeq1d 6777	. . . . . . . 8 ⊢ (𝜑 → (𝐹:(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃) ↔ ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃)))
32	29, 31	mpbird 257	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))
33		f1of 6782	. . . . . . 7 ⊢ (𝐹:(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃) → 𝐹:(𝐶 Func 𝐷)⟶(𝑂 Func 𝑃))
34	32, 33	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝐶 Func 𝐷)⟶(𝑂 Func 𝑃))
35		eqid 2737	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))) = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥)))
36		ovex 7401	. . . . . . . . 9 ⊢ (𝑦𝑁𝑥) ∈ V
37		resiexg 7864	. . . . . . . . 9 ⊢ ((𝑦𝑁𝑥) ∈ V → ( I ↾ (𝑦𝑁𝑥)) ∈ V)
38	36, 37	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ (𝑦𝑁𝑥)) ∈ V
39	35, 38	fnmpoi 8024	. . . . . . 7 ⊢ (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))) Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷))
40		fucoppc.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
41	40	fneq1d 6593	. . . . . . 7 ⊢ (𝜑 → (𝐺 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ↔ (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))) Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷))))
42	39, 41	mpbiri 258	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
43		f1oi 6820	. . . . . . . 8 ⊢ ( I ↾ (𝑔𝑁𝑓)):(𝑔𝑁𝑓)–1-1-onto→(𝑔𝑁𝑓)
44	40	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
45		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝑓 ∈ (𝐶 Func 𝐷))
46		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝑔 ∈ (𝐶 Func 𝐷))
47	44, 45, 46	opf2fval 49764	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → (𝑓𝐺𝑔) = ( I ↾ (𝑔𝑁𝑓)))
48		fucoppc.n	. . . . . . . . . . . 12 ⊢ 𝑁 = (𝐶 Nat 𝐷)
49	2, 48	fuchom 17900	. . . . . . . . . . 11 ⊢ 𝑁 = (Hom ‘𝑄)
50	49, 1	oppchom 17650	. . . . . . . . . 10 ⊢ (𝑓(Hom ‘𝑅)𝑔) = (𝑔𝑁𝑓)
51	50	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → (𝑓(Hom ‘𝑅)𝑔) = (𝑔𝑁𝑓))
52	30	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
53	25, 26, 48, 52, 45, 46	fucoppclem 49766	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → (𝑔𝑁𝑓) = ((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)))
54	53	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → ((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)) = (𝑔𝑁𝑓))
55	47, 51, 54	f1oeq123d 6776	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → ((𝑓𝐺𝑔):(𝑓(Hom ‘𝑅)𝑔)–1-1-onto→((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)) ↔ ( I ↾ (𝑔𝑁𝑓)):(𝑔𝑁𝑓)–1-1-onto→(𝑔𝑁𝑓)))
56	43, 55	mpbiri 258	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → (𝑓𝐺𝑔):(𝑓(Hom ‘𝑅)𝑔)–1-1-onto→((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)))
57		f1of 6782	. . . . . . 7 ⊢ ((𝑓𝐺𝑔):(𝑓(Hom ‘𝑅)𝑔)–1-1-onto→((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)) → (𝑓𝐺𝑔):(𝑓(Hom ‘𝑅)𝑔)⟶((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)))
58	56, 57	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → (𝑓𝐺𝑔):(𝑓(Hom ‘𝑅)𝑔)⟶((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)))
59	30	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
60	40	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
61		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝑓 ∈ (𝐶 Func 𝐷))
62	25, 26, 2, 1, 5, 48, 59, 60, 61	fucoppcid 49767	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → ((𝑓𝐺𝑓)‘((Id‘𝑅)‘𝑓)) = ((Id‘𝑆)‘(𝐹‘𝑓)))
63	30	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑘 ∈ (𝐶 Func 𝐷)) ∧ (𝑎 ∈ (𝑓(Hom ‘𝑅)𝑔) ∧ 𝑏 ∈ (𝑔(Hom ‘𝑅)𝑘))) → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
64	40	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑘 ∈ (𝐶 Func 𝐷)) ∧ (𝑎 ∈ (𝑓(Hom ‘𝑅)𝑔) ∧ 𝑏 ∈ (𝑔(Hom ‘𝑅)𝑘))) → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
65		simp3l 1203	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑘 ∈ (𝐶 Func 𝐷)) ∧ (𝑎 ∈ (𝑓(Hom ‘𝑅)𝑔) ∧ 𝑏 ∈ (𝑔(Hom ‘𝑅)𝑘))) → 𝑎 ∈ (𝑓(Hom ‘𝑅)𝑔))
66		simp3r 1204	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑘 ∈ (𝐶 Func 𝐷)) ∧ (𝑎 ∈ (𝑓(Hom ‘𝑅)𝑔) ∧ 𝑏 ∈ (𝑔(Hom ‘𝑅)𝑘))) → 𝑏 ∈ (𝑔(Hom ‘𝑅)𝑘))
67	25, 26, 2, 1, 5, 48, 63, 64, 65, 66	fucoppcco 49768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑘 ∈ (𝐶 Func 𝐷)) ∧ (𝑎 ∈ (𝑓(Hom ‘𝑅)𝑔) ∧ 𝑏 ∈ (𝑔(Hom ‘𝑅)𝑘))) → ((𝑓𝐺𝑘)‘(𝑏(⟨𝑓, 𝑔⟩(comp‘𝑅)𝑘)𝑎)) = (((𝑔𝐺𝑘)‘𝑏)(⟨(𝐹‘𝑓), (𝐹‘𝑔)⟩(comp‘𝑆)(𝐹‘𝑘))((𝑓𝐺𝑔)‘𝑎)))
68	4, 6, 7, 9, 10, 11, 12, 13, 21, 24, 34, 42, 58, 62, 67	isfuncd 17801	. . . . 5 ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)
69	56	ralrimivva 3181	. . . . 5 ⊢ (𝜑 → ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑔 ∈ (𝐶 Func 𝐷)(𝑓𝐺𝑔):(𝑓(Hom ‘𝑅)𝑔)–1-1-onto→((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)))
70	4, 7, 9	isffth2 17854	. . . . 5 ⊢ (𝐹((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆))𝐺 ↔ (𝐹(𝑅 Func 𝑆)𝐺 ∧ ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑔 ∈ (𝐶 Func 𝐷)(𝑓𝐺𝑔):(𝑓(Hom ‘𝑅)𝑔)–1-1-onto→((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔))))
71	68, 69, 70	sylanbrc 584	. . . 4 ⊢ (𝜑 → 𝐹((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆))𝐺)
72		df-br 5101	. . . 4 ⊢ (𝐹((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆))𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ ((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆)))
73	71, 72	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆)))
74	68	func1st 49436	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
75	74	f1oeq1d 6777	. . . 4 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃) ↔ 𝐹:(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃)))
76	32, 75	mpbird 257	. . 3 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))
77		fucoppc.i	. . . 4 ⊢ 𝐼 = (Iso‘𝑇)
78	15, 16, 4, 6, 18, 14, 22, 77	catciso 18047	. . 3 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆) ↔ (⟨𝐹, 𝐺⟩ ∈ ((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆)) ∧ (1st ‘⟨𝐹, 𝐺⟩):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))))
79	73, 76, 78	mpbir2and 714	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆))
80		df-br 5101	. 2 ⊢ (𝐹(𝑅𝐼𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆))
81	79, 80	sylibr 234	1 ⊢ (𝜑 → 𝐹(𝑅𝐼𝑆)𝐺)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ∩ cin 3902  ⟨cop 4588   class class class wbr 5100   I cid 5526   × cxp 5630   ↾ cres 5634   Fn wfn 6495  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  Basecbs 17148  Hom chom 17200  compcco 17201  Catccat 17599  Idccid 17600  oppCatcoppc 17646  Isociso 17682   Func cfunc 17790   Full cful 17840   Faith cfth 17841   Nat cnat 17880   FuncCat cfuc 17881  CatCatccatc 18034   oppFunc coppf 49481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-tpos 8178  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-hom 17213  df-cco 17214  df-cat 17603  df-cid 17604  df-homf 17605  df-comf 17606  df-oppc 17647  df-sect 17683  df-inv 17684  df-iso 17685  df-func 17794  df-idfu 17795  df-cofu 17796  df-full 17842  df-fth 17843  df-nat 17882  df-fuc 17883  df-catc 18035  df-oppf 49482

This theorem is referenced by:  fucoppcffth  49770  fucoppccic  49772