fucoppc - Mathbox for Zhi Wang

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Theorem fucoppc 50216

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 18026	. . . . . . 7 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
4	1, 3	oppcbas 17780	. . . . . 6 ⊢ (𝐶 Func 𝐷) = (Base‘𝑅)
5		fucoppc.s	. . . . . . 7 ⊢ 𝑆 = (𝑂 FuncCat 𝑃)
6	5	fucbas 18026	. . . . . 6 ⊢ (𝑂 Func 𝑃) = (Base‘𝑆)
7		eqid 2762	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
8		eqid 2762	. . . . . . 7 ⊢ (𝑂 Nat 𝑃) = (𝑂 Nat 𝑃)
9	5, 8	fuchom 18027	. . . . . 6 ⊢ (𝑂 Nat 𝑃) = (Hom ‘𝑆)
10		eqid 2762	. . . . . 6 ⊢ (Id‘𝑅) = (Id‘𝑅)
11		eqid 2762	. . . . . 6 ⊢ (Id‘𝑆) = (Id‘𝑆)
12		eqid 2762	. . . . . 6 ⊢ (comp‘𝑅) = (comp‘𝑅)
13		eqid 2762	. . . . . 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 17281	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝐵 → 𝑈 ∈ V)
18	14, 17	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
19	15, 16, 18	catcbas 18164	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
20	14, 19	eleqtrd 2864	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝑈 ∩ Cat))
21	20	elin2d 4157	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Cat)
22		fucoppc.2	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
23	22, 19	eleqtrd 2864	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (𝑈 ∩ Cat))
24	23	elin2d 4157	. . . . . 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 49955	. . . . . . . 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 6815	. . . . . . . 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 260	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))
33		f1of 6820	. . . . . . 7 ⊢ (𝐹:(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃) → 𝐹:(𝐶 Func 𝐷)⟶(𝑂 Func 𝑃))
34	32, 33	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝐶 Func 𝐷)⟶(𝑂 Func 𝑃))
35		eqid 2762	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))) = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥)))
36		ovex 7445	. . . . . . . . 9 ⊢ (𝑦𝑁𝑥) ∈ V
37		resiexg 7907	. . . . . . . . 9 ⊢ ((𝑦𝑁𝑥) ∈ V → ( I ↾ (𝑦𝑁𝑥)) ∈ V)
38	36, 37	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ (𝑦𝑁𝑥)) ∈ V
39	35, 38	fnmpoi 8065	. . . . . . 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 6628	. . . . . . 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 261	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
43		f1oi 6859	. . . . . . . 8 ⊢ ( I ↾ (𝑔𝑁𝑓)):(𝑔𝑁𝑓)–1-1-onto→(𝑔𝑁𝑓)
44	40	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
45		simprl 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝑓 ∈ (𝐶 Func 𝐷))
46		simprr 784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝑔 ∈ (𝐶 Func 𝐷))
47	44, 45, 46	opf2fval 50211	. . . . . . . . 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 18027	. . . . . . . . . . 11 ⊢ 𝑁 = (Hom ‘𝑄)
50	49, 1	oppchom 17777	. . . . . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
53	25, 26, 48, 52, 45, 46	fucoppclem 50213	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → (𝑔𝑁𝑓) = ((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)))
54	53	eqcomd 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → ((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)) = (𝑔𝑁𝑓))
55	47, 51, 54	f1oeq123d 6814	. . . . . . . 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 261	. . . . . . 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 6820	. . . . . . 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 18	. . . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
60	40	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
61		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝑓 ∈ (𝐶 Func 𝐷))
62	25, 26, 2, 1, 5, 48, 59, 60, 61	fucoppcid 50214	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → ((𝑓𝐺𝑓)‘((Id‘𝑅)‘𝑓)) = ((Id‘𝑆)‘(𝐹‘𝑓)))
63	30	3ad2ant1 1150	. . . . . . 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 1150	. . . . . . 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 1219	. . . . . . 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 1220	. . . . . . 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 50215	. . . . . 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 17928	. . . . 5 ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)
69	56	ralrimivva 3207	. . . . 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 17981	. . . . 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 594	. . . 4 ⊢ (𝜑 → 𝐹((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆))𝐺)
72		df-br 5109	. . . 4 ⊢ (𝐹((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆))𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ ((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆)))
73	71, 72	sylib 221	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆)))
74	68	func1st 49883	. . . . 5 ⊢ (𝜑 → (1^st ‘⟨𝐹, 𝐺⟩) = 𝐹)
75	74	f1oeq1d 6815	. . . 4 ⊢ (𝜑 → ((1^st ‘⟨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 260	. . 3 ⊢ (𝜑 → (1^st ‘⟨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 18174	. . 3 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆) ↔ (⟨𝐹, 𝐺⟩ ∈ ((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆)) ∧ (1^st ‘⟨𝐹, 𝐺⟩):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))))
79	73, 76, 78	mpbir2and 725	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆))
80		df-br 5109	. 2 ⊢ (𝐹(𝑅𝐼𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆))
81	79, 80	sylibr 237	1 ⊢ (𝜑 → 𝐹(𝑅𝐼𝑆)𝐺)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ∀wral 3078 Vcvv 3454 ∩ cin 3903 ⟨cop 4594 class class class wbr 5108 I cid 5554 × cxp 5658 ↾ cres 5662 Fn wfn 6531 ⟶wf 6532 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7412 ∈ cmpo 7414 1^st c1st 7982 Basecbs 17275 Hom chom 17327 compcco 17328 Catccat 17726 Idccid 17727 oppCatcoppc 17773 Isociso 17809 Func cfunc 17917 Full cful 17967 Faith cfth 17968 Nat cnat 18007 FuncCat cfuc 18008 CatCatccatc 18161 oppFunc coppf 49928

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-er 8692 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-fz 13542 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-hom 17340 df-cco 17341 df-cat 17730 df-cid 17731 df-homf 17732 df-comf 17733 df-oppc 17774 df-sect 17810 df-inv 17811 df-iso 17812 df-func 17921 df-idfu 17922 df-cofu 17923 df-full 17969 df-fth 17970 df-nat 18009 df-fuc 18010 df-catc 18162 df-oppf 49929

This theorem is used by: fucoppcffth 50217 fucoppccic 50219

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