Theorem funcoppc 18059

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 2760	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17901	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
4		funcoppc.p	. . 3 ⊢ 𝑃 = (oppCat‘𝐷)
5		eqid 2760	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
6	4, 5	oppcbas 17901	. 2 ⊢ (Base‘𝐷) = (Base‘𝑃)
7		eqid 2760	. 2 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
8		eqid 2760	. 2 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
9		eqid 2760	. 2 ⊢ (Id‘𝑂) = (Id‘𝑂)
10		eqid 2760	. 2 ⊢ (Id‘𝑃) = (Id‘𝑃)
11		eqid 2760	. 2 ⊢ (comp‘𝑂) = (comp‘𝑂)
12		eqid 2760	. 2 ⊢ (comp‘𝑃) = (comp‘𝑃)
13		funcoppc.f	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
14		df-br 5103	. . . . . 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 18047	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
17	15, 16	syl 18	. . . 4 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
18	17	simpld 500	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
19	1	oppccat 17905	. . 3 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
20	18, 19	syl 18	. 2 ⊢ (𝜑 → 𝑂 ∈ Cat)
21	4	oppccat 17905	. . 3 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
22	17, 21	simpl2im 513	. 2 ⊢ (𝜑 → 𝑃 ∈ Cat)
23	2, 5, 13	funcf1 18050	. 2 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
24	2, 13	funcfn2 18053	. . 3 ⊢ (𝜑 → 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
25		tposfn 8280	. . 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 2760	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
28		eqid 2760	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
29	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Func 𝐷)𝐺)
30		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
31		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
32	2, 27, 28, 29, 30, 31	funcf2 18052	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)))
33		ovtpos 8266	. . . . 5 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥)
34	33	feq1i 6705	. . . 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 17898	. . . . 5 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
36	28, 4	oppchom 17898	. . . . 5 ⊢ ((𝐹‘𝑥)(Hom ‘𝑃)(𝐹‘𝑦)) = ((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥))
37	35, 36	feq23i 6708	. . . 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 2760	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
41		eqid 2760	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
42	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹(𝐶 Func 𝐷)𝐺)
43		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
44	2, 40, 41, 42, 43	funcid 18054	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)))
45		ovtpos 8266	. . . . 5 ⊢ (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥)
46	45	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥))
47	1, 40	oppcid 17904	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
48	18, 47	syl 18	. . . . . 6 ⊢ (𝜑 → (Id‘𝑂) = (Id‘𝐶))
49	48	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (Id‘𝑂) = (Id‘𝐶))
50	49	fveq1d 6892	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑂)‘𝑥) = ((Id‘𝐶)‘𝑥))
51	46, 50	fveq12d 6897	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)))
52	4, 41	oppcid 17904	. . . . . 6 ⊢ (𝐷 ∈ Cat → (Id‘𝑃) = (Id‘𝐷))
53	17, 52	simpl2im 513	. . . . 5 ⊢ (𝜑 → (Id‘𝑃) = (Id‘𝐷))
54	53	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (Id‘𝑃) = (Id‘𝐷))
55	54	fveq1d 6892	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑃)‘(𝐹‘𝑥)) = ((Id‘𝐷)‘(𝐹‘𝑥)))
56	44, 51, 55	3eqtr4d 2805	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((Id‘𝑃)‘(𝐹‘𝑥)))
57		eqid 2760	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
58		eqid 2760	. . . . 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 17898	. . . . . 6 ⊢ (𝑦(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑦)
65	63, 64	eleqtrdi 2870	. . . . 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 2870	. . . . 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 18055	. . . 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 17900	. . . . 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 6894	. . . 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 7090	. . . . 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 7090	. . . . 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 7090	. . . . 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 17900	. . . 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 2805	. . 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 8266	. . . 4 ⊢ (𝑥tpos 𝐺𝑧) = (𝑧𝐺𝑥)
78	77	fveq1i 6891	. . 3 ⊢ ((𝑥tpos 𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = ((𝑧𝐺𝑥)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓))
79		ovtpos 8266	. . . . 5 ⊢ (𝑦tpos 𝐺𝑧) = (𝑧𝐺𝑦)
80	79	fveq1i 6891	. . . 4 ⊢ ((𝑦tpos 𝐺𝑧)‘𝑔) = ((𝑧𝐺𝑦)‘𝑔)
81	33	fveq1i 6891	. . . 4 ⊢ ((𝑥tpos 𝐺𝑦)‘𝑓) = ((𝑦𝐺𝑥)‘𝑓)
82	80, 81	oveq12i 7436	. . 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 2820	. 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 18049	1 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ⟨cop 4834   class class class wbr 5102   × cxp 5649   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424  tpos ctpos 8250  Basecbs 17396  Hom chom 17448  compcco 17449  Catccat 17847  Idccid 17848  oppCatcoppc 17894   Func cfunc 18038

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 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

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-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-tpos 8251  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-er 8725  df-map 8857  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-3 12413  df-4 12414  df-5 12415  df-6 12416  df-7 12417  df-8 12418  df-9 12419  df-n0 12614  df-z 12701  df-dec 12822  df-sets 17351  df-slot 17369  df-ndx 17381  df-base 17397  df-hom 17461  df-cco 17462  df-cat 17851  df-cid 17852  df-oppc 17895  df-func 18042

This theorem is used by:  fulloppc  18108  fthoppc  18109  yonedalem1  18455  yonedalem21  18456  yonedalem22  18461  oppfoppc  50393  funcoppc2  50395  cofuoppf  50402  oppcup  50459  natoppf  50487