Theorem fucoppc 50652

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 18145	. . . . . . 7 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
4	1, 3	oppcbas 17899	. . . . . 6 ⊢ (𝐶 Func 𝐷) = (Base‘𝑅)
5		fucoppc.s	. . . . . . 7 ⊢ 𝑆 = (𝑂 FuncCat 𝑃)
6	5	fucbas 18145	. . . . . 6 ⊢ (𝑂 Func 𝑃) = (Base‘𝑆)
7		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
8		eqid 2760	. . . . . . 7 ⊢ (𝑂 Nat 𝑃) = (𝑂 Nat 𝑃)
9	5, 8	fuchom 18146	. . . . . 6 ⊢ (𝑂 Nat 𝑃) = (Hom ‘𝑆)
10		eqid 2760	. . . . . 6 ⊢ (Id‘𝑅) = (Id‘𝑅)
11		eqid 2760	. . . . . 6 ⊢ (Id‘𝑆) = (Id‘𝑆)
12		eqid 2760	. . . . . 6 ⊢ (comp‘𝑅) = (comp‘𝑅)
13		eqid 2760	. . . . . 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 17400	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝐵 → 𝑈 ∈ V)
18	14, 17	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
19	15, 16, 18	catcbas 18283	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
20	14, 19	eleqtrd 2862	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝑈 ∩ Cat))
21	20	elin2d 4101	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Cat)
22		fucoppc.2	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
23	22, 19	eleqtrd 2862	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (𝑈 ∩ Cat))
24	23	elin2d 4101	. . . . . 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 50372	. . . . . . . 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 6822	. . . . . . . 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 6827	. . . . . . 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 2760	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))) = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥)))
36		ovex 7455	. . . . . . . . 9 ⊢ (𝑦𝑁𝑥) ∈ V
37		resiexg 7926	. . . . . . . . 9 ⊢ ((𝑦𝑁𝑥) ∈ V → ( I ↾ (𝑦𝑁𝑥)) ∈ V)
38	36, 37	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ (𝑦𝑁𝑥)) ∈ V
39	35, 38	fnmpoi 8083	. . . . . . 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 6635	. . . . . . 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 6866	. . . . . . . 8 ⊢ ( I ↾ (𝑔𝑁𝑓)):(𝑔𝑁𝑓)–1-1-onto→(𝑔𝑁𝑓)
44	40	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
45		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝑓 ∈ (𝐶 Func 𝐷))
46		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝑔 ∈ (𝐶 Func 𝐷))
47	44, 45, 46	opf2fval 50647	. . . . . . . . 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 18146	. . . . . . . . . . 11 ⊢ 𝑁 = (Hom ‘𝑄)
50	49, 1	oppchom 17896	. . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
53	25, 26, 48, 52, 45, 46	fucoppclem 50649	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → (𝑔𝑁𝑓) = ((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)))
54	53	eqcomd 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷))) → ((𝐹‘𝑓)(𝑂 Nat 𝑃)(𝐹‘𝑔)) = (𝑔𝑁𝑓))
55	47, 51, 54	f1oeq123d 6821	. . . . . . . 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 6827	. . . . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
60	40	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
61		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝑓 ∈ (𝐶 Func 𝐷))
62	25, 26, 2, 1, 5, 48, 59, 60, 61	fucoppcid 50650	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → ((𝑓𝐺𝑓)‘((Id‘𝑅)‘𝑓)) = ((Id‘𝑆)‘(𝐹‘𝑓)))
63	30	3ad2ant1 1151	. . . . . . 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 1151	. . . . . . 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 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))) → 𝑎 ∈ (𝑓(Hom ‘𝑅)𝑔))
66		simp3r 1221	. . . . . . 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 50651	. . . . . 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 18047	. . . . 5 ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)
69	56	ralrimivva 3205	. . . . 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 18100	. . . . 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 595	. . . 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 221	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆)))
74	68	func1st 50298	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
75	74	f1oeq1d 6822	. . . 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 260	. . 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 18293	. . 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 726	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆))
80		df-br 5103	. 2 ⊢ (𝐹(𝑅𝐼𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅𝐼𝑆))
81	79, 80	sylibr 237	1 ⊢ (𝜑 → 𝐹(𝑅𝐼𝑆)𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ∩ cin 4088  ⟨cop 4834   class class class wbr 5102   I cid 5544   × cxp 5648   ↾ cres 5652   Fn wfn 6536  ⟶wf 6537  –1-1-onto→wf1o 6540  ‘cfv 6541  (class class class)co 7422   ∈ cmpo 7424  1^st c1st 8001  Basecbs 17394  Hom chom 17446  compcco 17447  Catccat 17845  Idccid 17846  oppCatcoppc 17892  Isociso 17928   Func cfunc 18036   Full cful 18086   Faith cfth 18087   Nat cnat 18126   FuncCat cfuc 18127  CatCatccatc 18280   oppFunc coppf 50345

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-tpos 8248  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-er 8722  df-map 8854  df-ixp 8931  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-nn 12341  df-2 12410  df-3 12411  df-4 12412  df-5 12413  df-6 12414  df-7 12415  df-8 12416  df-9 12417  df-n0 12612  df-z 12699  df-dec 12820  df-uz 12971  df-fz 13645  df-struct 17332  df-sets 17349  df-slot 17367  df-ndx 17379  df-base 17395  df-hom 17459  df-cco 17460  df-cat 17849  df-cid 17850  df-homf 17851  df-comf 17852  df-oppc 17893  df-sect 17929  df-inv 17930  df-iso 17931  df-func 18040  df-idfu 18041  df-cofu 18042  df-full 18088  df-fth 18089  df-nat 18128  df-fuc 18129  df-catc 18281  df-oppf 50346

This theorem is used by:  fucoppcffth  50653  fucoppccic  50655