Theorem funcoppc 17927

Description: A functor on categories yields a functor on the opposite categories (in the same direction), see definition 3.41 of [Adamek] p. 39. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
funcoppc.o	⊢ 𝑂 = (oppCat‘𝐶)
funcoppc.p	⊢ 𝑃 = (oppCat‘𝐷)
funcoppc.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)

Assertion

Ref	Expression
funcoppc	⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)

Proof of Theorem funcoppc

Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcoppc.o	. . 3 ⊢ 𝑂 = (oppCat‘𝐶)
2		eqid 2763	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17769	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
4		funcoppc.p	. . 3 ⊢ 𝑃 = (oppCat‘𝐷)
5		eqid 2763	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
6	4, 5	oppcbas 17769	. 2 ⊢ (Base‘𝐷) = (Base‘𝑃)
7		eqid 2763	. 2 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
8		eqid 2763	. 2 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
9		eqid 2763	. 2 ⊢ (Id‘𝑂) = (Id‘𝑂)
10		eqid 2763	. 2 ⊢ (Id‘𝑃) = (Id‘𝑃)
11		eqid 2763	. 2 ⊢ (comp‘𝑂) = (comp‘𝑂)
12		eqid 2763	. 2 ⊢ (comp‘𝑃) = (comp‘𝑃)
13		funcoppc.f	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
14		df-br 5110	. . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
15	13, 14	sylib 221	. . . . 5 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
16		funcrcl 17915	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
17	15, 16	syl 18	. . . 4 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
18	17	simpld 499	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
19	1	oppccat 17773	. . 3 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
20	18, 19	syl 18	. 2 ⊢ (𝜑 → 𝑂 ∈ Cat)
21	4	oppccat 17773	. . 3 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
22	17, 21	simpl2im 512	. 2 ⊢ (𝜑 → 𝑃 ∈ Cat)
23	2, 5, 13	funcf1 17918	. 2 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
24	2, 13	funcfn2 17921	. . 3 ⊢ (𝜑 → 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
25		tposfn 8247	. . 3 ⊢ (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) → tpos 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
26	24, 25	syl 18	. 2 ⊢ (𝜑 → tpos 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
27		eqid 2763	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
28		eqid 2763	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
29	13	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Func 𝐷)𝐺)
30		simprr 784	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
31		simprl 782	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
32	2, 27, 28, 29, 30, 31	funcf2 17920	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)))
33		ovtpos 8233	. . . . 5 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥)
34	33	feq1i 6696	. . . 4 ⊢ ((𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)) ↔ (𝑦𝐺𝑥):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)))
35	27, 1	oppchom 17766	. . . . 5 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
36	28, 4	oppchom 17766	. . . . 5 ⊢ ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)) = ((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥))
37	35, 36	feq23i 6699	. . . 4 ⊢ ((𝑦𝐺𝑥):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)) ↔ (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)))
38	34, 37	bitri 278	. . 3 ⊢ ((𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)) ↔ (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)))
39	32, 38	sylibr 237	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)))
40		eqid 2763	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
41		eqid 2763	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
42	13	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹(𝐶 Func 𝐷)𝐺)
43		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
44	2, 40, 41, 42, 43	funcid 17922	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)))
45		ovtpos 8233	. . . . 5 ⊢ (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥)
46	45	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥))
47	1, 40	oppcid 17772	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
48	18, 47	syl 18	. . . . . 6 ⊢ (𝜑 → (Id‘𝑂) = (Id‘𝐶))
49	48	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (Id‘𝑂) = (Id‘𝐶))
50	49	fveq1d 6883	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑂)‘𝑥) = ((Id‘𝐶)‘𝑥))
51	46, 50	fveq12d 6888	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)))
52	4, 41	oppcid 17772	. . . . . 6 ⊢ (𝐷 ∈ Cat → (Id‘𝑃) = (Id‘𝐷))
53	17, 52	simpl2im 512	. . . . 5 ⊢ (𝜑 → (Id‘𝑃) = (Id‘𝐷))
54	53	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (Id‘𝑃) = (Id‘𝐷))
55	54	fveq1d 6883	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑃)‘(𝐹‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)))
56	44, 51, 55	3eqtr4d 2808	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((Id‘𝑃)‘(𝐹‘𝑥)))
57		eqid 2763	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
58		eqid 2763	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
59	13	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝐹(𝐶 Func 𝐷)𝐺)
60		simp23 1227	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑧 ∈ (Base‘𝐶))
61		simp22 1226	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑦 ∈ (Base‘𝐶))
62		simp21 1225	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑥 ∈ (Base‘𝐶))
63		simp3r 1221	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))
64	27, 1	oppchom 17766	. . . . . 6 ⊢ (𝑦(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑦)
65	63, 64	eleqtrdi 2873	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑦))
66		simp3l 1220	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦))
67	66, 35	eleqtrdi 2873	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))
68	2, 27, 57, 58, 59, 60, 61, 62, 65, 67	funcco 17923	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → ((𝑧𝐺𝑥)‘(𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔)) = (((𝑦𝐺𝑥)‘𝑓)(⟨(𝐹‘𝑧), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑥))((𝑧𝐺𝑦)‘𝑔)))
69	2, 57, 1, 62, 61, 60	oppcco 17768	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓) = (𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔))
70	69	fveq2d 6885	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → ((𝑧𝐺𝑥)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = ((𝑧𝐺𝑥)‘(𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔)))
71	23	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
72	71, 62	ffvelcdmd 7080	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (𝐹‘𝑥) ∈ (Base‘𝐷))
73	71, 61	ffvelcdmd 7080	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (𝐹‘𝑦) ∈ (Base‘𝐷))
74	71, 60	ffvelcdmd 7080	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (𝐹‘𝑧) ∈ (Base‘𝐷))
75	5, 58, 4, 72, 73, 74	oppcco 17768	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (((𝑧𝐺𝑦)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝑃)(𝐹‘𝑧))((𝑦𝐺𝑥)‘𝑓)) = (((𝑦𝐺𝑥)‘𝑓)(⟨(𝐹‘𝑧), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑥))((𝑧𝐺𝑦)‘𝑔)))
76	68, 70, 75	3eqtr4d 2808	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → ((𝑧𝐺𝑥)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = (((𝑧𝐺𝑦)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝑃)(𝐹‘𝑧))((𝑦𝐺𝑥)‘𝑓)))
77		ovtpos 8233	. . . 4 ⊢ (𝑥tpos 𝐺𝑧) = (𝑧𝐺𝑥)
78	77	fveq1i 6882	. . 3 ⊢ ((𝑥tpos 𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = ((𝑧𝐺𝑥)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓))
79		ovtpos 8233	. . . . 5 ⊢ (𝑦tpos 𝐺𝑧) = (𝑧𝐺𝑦)
80	79	fveq1i 6882	. . . 4 ⊢ ((𝑦tpos 𝐺𝑧)‘𝑔) = ((𝑧𝐺𝑦)‘𝑔)
81	33	fveq1i 6882	. . . 4 ⊢ ((𝑥tpos 𝐺𝑦)‘𝑓) = ((𝑦𝐺𝑥)‘𝑓)
82	80, 81	oveq12i 7422	. . 3 ⊢ (((𝑦tpos 𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝑃)(𝐹‘𝑧))((𝑥tpos 𝐺𝑦)‘𝑓)) = (((𝑧𝐺𝑦)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝑃)(𝐹‘𝑧))((𝑦𝐺𝑥)‘𝑓))
83	76, 78, 82	3eqtr4g 2823	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → ((𝑥tpos 𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = (((𝑦tpos 𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝑃)(𝐹‘𝑧))((𝑥tpos 𝐺𝑦)‘𝑓)))
84	3, 6, 7, 8, 9, 10, 11, 12, 20, 22, 23, 26, 39, 56, 83	isfuncd 17917	1 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ⟨cop 4595   class class class wbr 5109   × cxp 5659   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  tpos ctpos 8217  Basecbs 17264  Hom chom 17316  compcco 17317  Catccat 17715  Idccid 17716  oppCatcoppc 17762   Func cfunc 17906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  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 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-hom 17329  df-cco 17330  df-cat 17719  df-cid 17720  df-oppc 17763  df-func 17910

This theorem is referenced by:  fulloppc  17976  fthoppc  17977  yonedalem1  18323  yonedalem21  18324  yonedalem22  18329  oppfoppc  49919  funcoppc2  49921  cofuoppf  49928  oppcup  49985  natoppf  50007