Theorem funcoppc 17921

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 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17762	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
4		funcoppc.p	. . 3 ⊢ 𝑃 = (oppCat‘𝐷)
5		eqid 2736	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
6	4, 5	oppcbas 17762	. 2 ⊢ (Base‘𝐷) = (Base‘𝑃)
7		eqid 2736	. 2 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
8		eqid 2736	. 2 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
9		eqid 2736	. 2 ⊢ (Id‘𝑂) = (Id‘𝑂)
10		eqid 2736	. 2 ⊢ (Id‘𝑃) = (Id‘𝑃)
11		eqid 2736	. 2 ⊢ (comp‘𝑂) = (comp‘𝑂)
12		eqid 2736	. 2 ⊢ (comp‘𝑃) = (comp‘𝑃)
13		funcoppc.f	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
14		df-br 5143	. . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
15	13, 14	sylib 218	. . . . 5 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
16		funcrcl 17909	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
17	15, 16	syl 17	. . . 4 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
18	17	simpld 494	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
19	1	oppccat 17766	. . 3 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
20	18, 19	syl 17	. 2 ⊢ (𝜑 → 𝑂 ∈ Cat)
21	4	oppccat 17766	. . 3 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
22	17, 21	simpl2im 503	. 2 ⊢ (𝜑 → 𝑃 ∈ Cat)
23	2, 5, 13	funcf1 17912	. 2 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
24	2, 13	funcfn2 17915	. . 3 ⊢ (𝜑 → 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
25		tposfn 8281	. . 3 ⊢ (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) → tpos 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
26	24, 25	syl 17	. 2 ⊢ (𝜑 → tpos 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
27		eqid 2736	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
28		eqid 2736	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
29	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Func 𝐷)𝐺)
30		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
31		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
32	2, 27, 28, 29, 30, 31	funcf2 17914	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)))
33		ovtpos 8267	. . . . 5 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥)
34	33	feq1i 6726	. . . 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 17759	. . . . 5 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
36	28, 4	oppchom 17759	. . . . 5 ⊢ ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)) = ((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥))
37	35, 36	feq23i 6729	. . . 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 275	. . 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 234	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)))
40		eqid 2736	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
41		eqid 2736	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
42	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹(𝐶 Func 𝐷)𝐺)
43		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
44	2, 40, 41, 42, 43	funcid 17916	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)))
45		ovtpos 8267	. . . . 5 ⊢ (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥)
46	45	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥))
47	1, 40	oppcid 17765	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
48	18, 47	syl 17	. . . . . 6 ⊢ (𝜑 → (Id‘𝑂) = (Id‘𝐶))
49	48	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (Id‘𝑂) = (Id‘𝐶))
50	49	fveq1d 6907	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑂)‘𝑥) = ((Id‘𝐶)‘𝑥))
51	46, 50	fveq12d 6912	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)))
52	4, 41	oppcid 17765	. . . . . 6 ⊢ (𝐷 ∈ Cat → (Id‘𝑃) = (Id‘𝐷))
53	17, 52	simpl2im 503	. . . . 5 ⊢ (𝜑 → (Id‘𝑃) = (Id‘𝐷))
54	53	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (Id‘𝑃) = (Id‘𝐷))
55	54	fveq1d 6907	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑃)‘(𝐹‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)))
56	44, 51, 55	3eqtr4d 2786	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((Id‘𝑃)‘(𝐹‘𝑥)))
57		eqid 2736	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
58		eqid 2736	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
59	13	3ad2ant1 1133	. . . . 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 1208	. . . . 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 1207	. . . . 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 1206	. . . . 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 1202	. . . . . 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 17759	. . . . . 6 ⊢ (𝑦(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑦)
65	63, 64	eleqtrdi 2850	. . . . 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 1201	. . . . . 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 2850	. . . . 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 17917	. . . 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 17761	. . . . 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 6909	. . . 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 1133	. . . . . 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 7104	. . . . 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 7104	. . . . 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 7104	. . . . 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 17761	. . . 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 2786	. . 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 8267	. . . 4 ⊢ (𝑥tpos 𝐺𝑧) = (𝑧𝐺𝑥)
78	77	fveq1i 6906	. . 3 ⊢ ((𝑥tpos 𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = ((𝑧𝐺𝑥)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓))
79		ovtpos 8267	. . . . 5 ⊢ (𝑦tpos 𝐺𝑧) = (𝑧𝐺𝑦)
80	79	fveq1i 6906	. . . 4 ⊢ ((𝑦tpos 𝐺𝑧)‘𝑔) = ((𝑧𝐺𝑦)‘𝑔)
81	33	fveq1i 6906	. . . 4 ⊢ ((𝑥tpos 𝐺𝑦)‘𝑓) = ((𝑦𝐺𝑥)‘𝑓)
82	80, 81	oveq12i 7444	. . 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 2801	. 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 17911	1 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ⟨cop 4631   class class class wbr 5142   × cxp 5682   Fn wfn 6555  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432  tpos ctpos 8251  Basecbs 17248  Hom chom 17309  compcco 17310  Catccat 17708  Idccid 17709  oppCatcoppc 17755   Func cfunc 17900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-hom 17322  df-cco 17323  df-cat 17712  df-cid 17713  df-oppc 17756  df-func 17904

This theorem is referenced by:  fulloppc  17970  fthoppc  17971  yonedalem1  18318  yonedalem21  18319  yonedalem22  18324  oppcup  48966