Theorem fucoppc 49992

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 17987	. . . . . . 7 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
4	1, 3	oppcbas 17741	. . . . . 6 ⊢ (𝐶 Func 𝐷) = (Base‘𝑅)
5		fucoppc.s	. . . . . . 7 ⊢ 𝑆 = (𝑂 FuncCat 𝑃)
6	5	fucbas 17987	. . . . . 6 ⊢ (𝑂 Func 𝑃) = (Base‘𝑆)
7		eqid 2761	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
8		eqid 2761	. . . . . . 7 ⊢ (𝑂 Nat 𝑃) = (𝑂 Nat 𝑃)
9	5, 8	fuchom 17988	. . . . . 6 ⊢ (𝑂 Nat 𝑃) = (Hom ‘𝑆)
10		eqid 2761	. . . . . 6 ⊢ (Id‘𝑅) = (Id‘𝑅)
11		eqid 2761	. . . . . 6 ⊢ (Id‘𝑆) = (Id‘𝑆)
12		eqid 2761	. . . . . 6 ⊢ (comp‘𝑅) = (comp‘𝑅)
13		eqid 2761	. . . . . 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 17242	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝐵 → 𝑈 ∈ V)
18	14, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
19	15, 16, 18	catcbas 18125	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
20	14, 19	eleqtrd 2863	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝑈 ∩ Cat))
21	20	elin2d 4155	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Cat)
22		fucoppc.2	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
23	22, 19	eleqtrd 2863	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (𝑈 ∩ Cat))
24	23	elin2d 4155	. . . . . 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 49731	. . . . . . . 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 6796	. . . . . . . 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 259	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))
33		f1of 6801	. . . . . . 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 2761	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))) = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥)))
36		ovex 7424	. . . . . . . . 9 ⊢ (𝑦𝑁𝑥) ∈ V
37		resiexg 7888	. . . . . . . . 9 ⊢ ((𝑦𝑁𝑥) ∈ V → ( I ↾ (𝑦𝑁𝑥)) ∈ V)
38	36, 37	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ (𝑦𝑁𝑥)) ∈ V
39	35, 38	fnmpoi 8046	. . . . . . 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 6609	. . . . . . 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 260	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
43		f1oi 6840	. . . . . . . 8 ⊢ ( I ↾ (𝑔𝑁𝑓)):(𝑔𝑁𝑓)–1-1-onto→(𝑔𝑁𝑓)
44	40	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
45		simprl 780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝑓 ∈ (𝐶 Func 𝐷))
46		simprr 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝑔 ∈ (𝐶 Func 𝐷))
47	44, 45, 46	opf2fval 49987	. . . . . . . . 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 17988	. . . . . . . . . . 11 ⊢ 𝑁 = (Hom ‘𝑄)
50	49, 1	oppchom 17738	. . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
53	25, 26, 48, 52, 45, 46	fucoppclem 49989	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → (𝑔𝑁𝑓) = ((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)))
54	53	eqcomd 2767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → ((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)) = (𝑔𝑁𝑓))
55	47, 51, 54	f1oeq123d 6795	. . . . . . . 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 260	. . . . . . 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 6801	. . . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
60	40	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
61		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝑓 ∈ (𝐶 Func 𝐷))
62	25, 26, 2, 1, 5, 48, 59, 60, 61	fucoppcid 49990	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → ((𝑓𝐺𝑓)‘((Id‘𝑅)‘𝑓)) = ((Id‘𝑆)‘(𝐹‘𝑓)))
63	30	3ad2ant1 1145	. . . . . . 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 1145	. . . . . . 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 1214	. . . . . . 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 1215	. . . . . . 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 49991	. . . . . 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 17889	. . . . 5 ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)
69	56	ralrimivva 3204	. . . . 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 17942	. . . . 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 592	. . . 4 ⊢ (𝜑 → 𝐹((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆))𝐺)
72		df-br 5098	. . . 4 ⊢ (𝐹((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆))𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ ((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆)))
73	71, 72	sylib 220	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆)))
74	68	func1st 49659	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
75	74	f1oeq1d 6796	. . . 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 259	. . 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 18135	. . 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 723	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆))
80		df-br 5098	. 2 ⊢ (𝐹(𝑅𝐼𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆))
81	79, 80	sylibr 236	1 ⊢ (𝜑 → 𝐹(𝑅𝐼𝑆)𝐺)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ∩ cin 3901  ⟨cop 4585   class class class wbr 5097   I cid 5537   × cxp 5641   ↾ cres 5645   Fn wfn 6511  ⟶wf 6512  –1-1-onto→wf1o 6515  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  1^st c1st 7963  Basecbs 17236  Hom chom 17288  compcco 17289  Catccat 17687  Idccid 17688  oppCatcoppc 17734  Isociso 17770   Func cfunc 17878   Full cful 17928   Faith cfth 17929   Nat cnat 17968   FuncCat cfuc 17969  CatCatccatc 18122   oppFunc coppf 49704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-tpos 8200  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-er 8672  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12476  df-z 12563  df-dec 12683  df-uz 12834  df-fz 13507  df-struct 17174  df-sets 17191  df-slot 17209  df-ndx 17221  df-base 17237  df-hom 17301  df-cco 17302  df-cat 17691  df-cid 17692  df-homf 17693  df-comf 17694  df-oppc 17735  df-sect 17771  df-inv 17772  df-iso 17773  df-func 17882  df-idfu 17883  df-cofu 17884  df-full 17930  df-fth 17931  df-nat 17970  df-fuc 17971  df-catc 18123  df-oppf 49705

This theorem is referenced by:  fucoppcffth  49993  fucoppccic  49995