Theorem fucoppc 49372

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 17901	. . . . . . 7 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
4	1, 3	oppcbas 17655	. . . . . 6 ⊢ (𝐶 Func 𝐷) = (Base‘𝑅)
5		fucoppc.s	. . . . . . 7 ⊢ 𝑆 = (𝑂 FuncCat 𝑃)
6	5	fucbas 17901	. . . . . 6 ⊢ (𝑂 Func 𝑃) = (Base‘𝑆)
7		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
8		eqid 2729	. . . . . . 7 ⊢ (𝑂 Nat 𝑃) = (𝑂 Nat 𝑃)
9	5, 8	fuchom 17902	. . . . . 6 ⊢ (𝑂 Nat 𝑃) = (Hom ‘𝑆)
10		eqid 2729	. . . . . 6 ⊢ (Id‘𝑅) = (Id‘𝑅)
11		eqid 2729	. . . . . 6 ⊢ (Id‘𝑆) = (Id‘𝑆)
12		eqid 2729	. . . . . 6 ⊢ (comp‘𝑅) = (comp‘𝑅)
13		eqid 2729	. . . . . 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 17161	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝐵 → 𝑈 ∈ V)
18	14, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
19	15, 16, 18	catcbas 18039	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
20	14, 19	eleqtrd 2830	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝑈 ∩ Cat))
21	20	elin2d 4164	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Cat)
22		fucoppc.2	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
23	22, 19	eleqtrd 2830	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (𝑈 ∩ Cat))
24	23	elin2d 4164	. . . . . 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 49111	. . . . . . . 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 2729	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))) = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥)))
36		ovex 7402	. . . . . . . . 9 ⊢ (𝑦𝑁𝑥) ∈ V
37		resiexg 7868	. . . . . . . . 9 ⊢ ((𝑦𝑁𝑥) ∈ V → ( I ↾ (𝑦𝑁𝑥)) ∈ V)
38	36, 37	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ (𝑦𝑁𝑥)) ∈ V
39	35, 38	fnmpoi 8028	. . . . . . 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 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝑓 ∈ (𝐶 Func 𝐷))
46		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝑔 ∈ (𝐶 Func 𝐷))
47	44, 45, 46	opf2fval 49367	. . . . . . . . 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 17902	. . . . . . . . . . 11 ⊢ 𝑁 = (Hom ‘𝑄)
50	49, 1	oppchom 17652	. . . . . . . . . 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 49369	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → (𝑔𝑁𝑓) = ((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)))
54	53	eqcomd 2735	. . . . . . . . 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 49370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → ((𝑓𝐺𝑓)‘((Id‘𝑅)‘𝑓)) = ((Id‘𝑆)‘(𝐹‘𝑓)))
63	30	3ad2ant1 1133	. . . . . . 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 1133	. . . . . . 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 1202	. . . . . . 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 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))) → 𝑏 ∈ (𝑔(Hom ‘𝑅)𝑘))
67	25, 26, 2, 1, 5, 48, 63, 64, 65, 66	fucoppcco 49371	. . . . . 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 17803	. . . . 5 ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)
69	56	ralrimivva 3178	. . . . 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 17856	. . . . 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 583	. . . 4 ⊢ (𝜑 → 𝐹((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆))𝐺)
72		df-br 5103	. . . 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 49039	. . . . 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 18049	. . 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 713	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆))
80		df-br 5103	. 2 ⊢ (𝐹(𝑅𝐼𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆))
81	79, 80	sylibr 234	1 ⊢ (𝜑 → 𝐹(𝑅𝐼𝑆)𝐺)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ∩ cin 3910  ⟨cop 4591   class class class wbr 5102   I cid 5525   × cxp 5629   ↾ cres 5633   Fn wfn 6494  ⟶wf 6495  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  1^st c1st 7945  Basecbs 17155  Hom chom 17207  compcco 17208  Catccat 17601  Idccid 17602  oppCatcoppc 17648  Isociso 17684   Func cfunc 17792   Full cful 17842   Faith cfth 17843   Nat cnat 17882   FuncCat cfuc 17883  CatCatccatc 18036   oppFunc coppf 49084

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-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-hom 17220  df-cco 17221  df-cat 17605  df-cid 17606  df-homf 17607  df-comf 17608  df-oppc 17649  df-sect 17685  df-inv 17686  df-iso 17687  df-func 17796  df-idfu 17797  df-cofu 17798  df-full 17844  df-fth 17845  df-nat 17884  df-fuc 17885  df-catc 18037  df-oppf 49085

This theorem is referenced by:  fucoppcffth  49373  fucoppccic  49375