Theorem xpcco 18350

Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
xpccofval.t	⊢ 𝑇 = (𝐶 ×c 𝐷)
xpccofval.b	⊢ 𝐵 = (Base‘𝑇)
xpccofval.k	⊢ 𝐾 = (Hom ‘𝑇)
xpccofval.o1	⊢ · = (comp‘𝐶)
xpccofval.o2	⊢ ∙ = (comp‘𝐷)
xpccofval.o	⊢ 𝑂 = (comp‘𝑇)
xpcco.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
xpcco.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
xpcco.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
xpcco.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))
xpcco.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))

Assertion

Ref	Expression
xpcco	⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩ · (1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ ∙ (2nd ‘𝑍))(2nd ‘𝐹))⟩)

Proof of Theorem xpcco

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

Step	Hyp	Ref	Expression
1		xpccofval.t	. . 3 ⊢ 𝑇 = (𝐶 ×c 𝐷)
2		xpccofval.b	. . 3 ⊢ 𝐵 = (Base‘𝑇)
3		xpccofval.k	. . 3 ⊢ 𝐾 = (Hom ‘𝑇)
4		xpccofval.o1	. . 3 ⊢ · = (comp‘𝐶)
5		xpccofval.o2	. . 3 ⊢ ∙ = (comp‘𝐷)
6		xpccofval.o	. . 3 ⊢ 𝑂 = (comp‘𝑇)
7	1, 2, 3, 4, 5, 6	xpccofval 18349	. 2 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩))
8		xpcco.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		xpcco.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10	8, 9	opelxpd 5690	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11		xpcco.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
12	11	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝑋, 𝑌⟩) → 𝑍 ∈ 𝐵)
13		ovex 7451	. . . . 5 ⊢ ((2nd ‘𝑥)𝐾𝑦) ∈ V
14		fvex 6896	. . . . 5 ⊢ (𝐾‘𝑥) ∈ V
15	13, 14	mpoex 8090	. . . 4 ⊢ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩) ∈ V
16	15	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩) ∈ V)
17		xpcco.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))
18	17	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐺 ∈ (𝑌𝐾𝑍))
19		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑥 = ⟨𝑋, 𝑌⟩)
20	19	fveq2d 6887	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd ‘𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
21		op2ndg 8012	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
22	8, 9, 21	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
23	22	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
24	20, 23	eqtrd 2796	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd ‘𝑥) = 𝑌)
25		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑦 = 𝑍)
26	24, 25	oveq12d 7436	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → ((2nd ‘𝑥)𝐾𝑦) = (𝑌𝐾𝑍))
27	18, 26	eleqtrrd 2864	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐺 ∈ ((2nd ‘𝑥)𝐾𝑦))
28		xpcco.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))
29	28	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐹 ∈ (𝑋𝐾𝑌))
30	19	fveq2d 6887	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾‘𝑥) = (𝐾‘⟨𝑋, 𝑌⟩))
31		df-ov 7421	. . . . . . 7 ⊢ (𝑋𝐾𝑌) = (𝐾‘⟨𝑋, 𝑌⟩)
32	30, 31	eqtr4di 2814	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾‘𝑥) = (𝑋𝐾𝑌))
33	29, 32	eleqtrrd 2864	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐹 ∈ (𝐾‘𝑥))
34	33	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ 𝑔 = 𝐺) → 𝐹 ∈ (𝐾‘𝑥))
35		opex 5432	. . . . 5 ⊢ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩ ∈ V
36	35	a1i 11	. . . 4 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩ ∈ V)
37	19	fveq2d 6887	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st ‘𝑥) = (1st ‘⟨𝑋, 𝑌⟩))
38		op1stg 8011	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
39	8, 9, 38	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
40	39	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
41	37, 40	eqtrd 2796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st ‘𝑥) = 𝑋)
42	41	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1st ‘𝑥) = 𝑋)
43	42	fveq2d 6887	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1st ‘(1st ‘𝑥)) = (1st ‘𝑋))
44	24	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑥) = 𝑌)
45	44	fveq2d 6887	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1st ‘(2nd ‘𝑥)) = (1st ‘𝑌))
46	43, 45	opeq12d 4841	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ = ⟨(1st ‘𝑋), (1st ‘𝑌)⟩)
47		simplrr 790	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑦 = 𝑍)
48	47	fveq2d 6887	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1st ‘𝑦) = (1st ‘𝑍))
49	46, 48	oveq12d 7436	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦)) = (⟨(1st ‘𝑋), (1st ‘𝑌)⟩ · (1st ‘𝑍)))
50		simprl 783	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
51	50	fveq2d 6887	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1st ‘𝑔) = (1st ‘𝐺))
52		simprr 785	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
53	52	fveq2d 6887	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1st ‘𝑓) = (1st ‘𝐹))
54	49, 51, 53	oveq123d 7439	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)) = ((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩ · (1st ‘𝑍))(1st ‘𝐹)))
55	42	fveq2d 6887	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘(1st ‘𝑥)) = (2nd ‘𝑋))
56	44	fveq2d 6887	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘(2nd ‘𝑥)) = (2nd ‘𝑌))
57	55, 56	opeq12d 4841	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ = ⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩)
58	47	fveq2d 6887	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑦) = (2nd ‘𝑍))
59	57, 58	oveq12d 7436	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦)) = (⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ ∙ (2nd ‘𝑍)))
60	50	fveq2d 6887	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑔) = (2nd ‘𝐺))
61	52	fveq2d 6887	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑓) = (2nd ‘𝐹))
62	59, 60, 61	oveq123d 7439	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓)) = ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ ∙ (2nd ‘𝑍))(2nd ‘𝐹)))
63	54, 62	opeq12d 4841	. . . 4 ⊢ (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩ = ⟨((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩ · (1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ ∙ (2nd ‘𝑍))(2nd ‘𝐹))⟩)
64	27, 34, 36, 63	ovmpodv2 7576	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → ((⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩) → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩ · (1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ ∙ (2nd ‘𝑍))(2nd ‘𝐹))⟩))
65	10, 12, 16, 64	ovmpodv 7575	. 2 ⊢ (𝜑 → (𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩)) → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩ · (1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ ∙ (2nd ‘𝑍))(2nd ‘𝐹))⟩))
66	7, 65	mpi 21	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩ · (1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ ∙ (2nd ‘𝑍))(2nd ‘𝐹))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590   × cxp 5649  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  1^st c1st 7997  2^nd c2nd 7998  Basecbs 17380  Hom chom 17432  compcco 17433   ×_c cxpc 18335

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-struct 17318  df-slot 17353  df-ndx 17365  df-base 17381  df-hom 17445  df-cco 17446  df-xpc 18339

This theorem is used by:  xpcco1st  18351  xpcco2nd  18352  xpcco2  18354  xpccatid  18355  swapfcoa  50358