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Definition df-xpc 17416

Description: Define the binary product of categories, which has objects for each pair of objects of the factors, and morphisms for each pair of morphisms of the factors. Composition is componentwise. (Contributed by Mario Carneiro, 10-Jan-2017.)

**Assertion**
Ref	Expression
df-xpc	⊢ ×_c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})

Distinct variable group: 𝑓,𝑏,𝑔,ℎ,𝑟,𝑠,𝑢,𝑣,𝑥,𝑦

**Detailed syntax breakdown of Definition df-xpc**
Step	Hyp	Ref	Expression
1		cxpc 17412	. 2 class ×_c
2		vr	. . 3 setvar 𝑟
3		vs	. . 3 setvar 𝑠
4		cvv 3494	. . 3 class V
5		vb	. . . 4 setvar 𝑏
6	2	cv 1532	. . . . . 6 class 𝑟
7		cbs 16477	. . . . . 6 class Base
8	6, 7	cfv 6349	. . . . 5 class (Base‘𝑟)
9	3	cv 1532	. . . . . 6 class 𝑠
10	9, 7	cfv 6349	. . . . 5 class (Base‘𝑠)
11	8, 10	cxp 5547	. . . 4 class ((Base‘𝑟) × (Base‘𝑠))
12		vh	. . . . 5 setvar ℎ
13		vu	. . . . . 6 setvar 𝑢
14		vv	. . . . . 6 setvar 𝑣
15	5	cv 1532	. . . . . 6 class 𝑏
16	13	cv 1532	. . . . . . . . 9 class 𝑢
17		c1st 7681	. . . . . . . . 9 class 1^st
18	16, 17	cfv 6349	. . . . . . . 8 class (1^st ‘𝑢)
19	14	cv 1532	. . . . . . . . 9 class 𝑣
20	19, 17	cfv 6349	. . . . . . . 8 class (1^st ‘𝑣)
21		chom 16570	. . . . . . . . 9 class Hom
22	6, 21	cfv 6349	. . . . . . . 8 class (Hom ‘𝑟)
23	18, 20, 22	co 7150	. . . . . . 7 class ((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣))
24		c2nd 7682	. . . . . . . . 9 class 2^nd
25	16, 24	cfv 6349	. . . . . . . 8 class (2^nd ‘𝑢)
26	19, 24	cfv 6349	. . . . . . . 8 class (2^nd ‘𝑣)
27	9, 21	cfv 6349	. . . . . . . 8 class (Hom ‘𝑠)
28	25, 26, 27	co 7150	. . . . . . 7 class ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣))
29	23, 28	cxp 5547	. . . . . 6 class (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))
30	13, 14, 15, 15, 29	cmpo 7152	. . . . 5 class (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣))))
31		cnx 16474	. . . . . . . 8 class ndx
32	31, 7	cfv 6349	. . . . . . 7 class (Base‘ndx)
33	32, 15	cop 4566	. . . . . 6 class ⟨(Base‘ndx), 𝑏⟩
34	31, 21	cfv 6349	. . . . . . 7 class (Hom ‘ndx)
35	12	cv 1532	. . . . . . 7 class ℎ
36	34, 35	cop 4566	. . . . . 6 class ⟨(Hom ‘ndx), ℎ⟩
37		cco 16571	. . . . . . . 8 class comp
38	31, 37	cfv 6349	. . . . . . 7 class (comp‘ndx)
39		vx	. . . . . . . 8 setvar 𝑥
40		vy	. . . . . . . 8 setvar 𝑦
41	15, 15	cxp 5547	. . . . . . . 8 class (𝑏 × 𝑏)
42		vg	. . . . . . . . 9 setvar 𝑔
43		vf	. . . . . . . . 9 setvar 𝑓
44	39	cv 1532	. . . . . . . . . . 11 class 𝑥
45	44, 24	cfv 6349	. . . . . . . . . 10 class (2^nd ‘𝑥)
46	40	cv 1532	. . . . . . . . . 10 class 𝑦
47	45, 46, 35	co 7150	. . . . . . . . 9 class ((2^nd ‘𝑥)ℎ𝑦)
48	44, 35	cfv 6349	. . . . . . . . 9 class (ℎ‘𝑥)
49	42	cv 1532	. . . . . . . . . . . 12 class 𝑔
50	49, 17	cfv 6349	. . . . . . . . . . 11 class (1^st ‘𝑔)
51	43	cv 1532	. . . . . . . . . . . 12 class 𝑓
52	51, 17	cfv 6349	. . . . . . . . . . 11 class (1^st ‘𝑓)
53	44, 17	cfv 6349	. . . . . . . . . . . . . 14 class (1^st ‘𝑥)
54	53, 17	cfv 6349	. . . . . . . . . . . . 13 class (1^st ‘(1^st ‘𝑥))
55	45, 17	cfv 6349	. . . . . . . . . . . . 13 class (1^st ‘(2^nd ‘𝑥))
56	54, 55	cop 4566	. . . . . . . . . . . 12 class ⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩
57	46, 17	cfv 6349	. . . . . . . . . . . 12 class (1^st ‘𝑦)
58	6, 37	cfv 6349	. . . . . . . . . . . 12 class (comp‘𝑟)
59	56, 57, 58	co 7150	. . . . . . . . . . 11 class (⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))
60	50, 52, 59	co 7150	. . . . . . . . . 10 class ((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓))
61	49, 24	cfv 6349	. . . . . . . . . . 11 class (2^nd ‘𝑔)
62	51, 24	cfv 6349	. . . . . . . . . . 11 class (2^nd ‘𝑓)
63	53, 24	cfv 6349	. . . . . . . . . . . . 13 class (2^nd ‘(1^st ‘𝑥))
64	45, 24	cfv 6349	. . . . . . . . . . . . 13 class (2^nd ‘(2^nd ‘𝑥))
65	63, 64	cop 4566	. . . . . . . . . . . 12 class ⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩
66	46, 24	cfv 6349	. . . . . . . . . . . 12 class (2^nd ‘𝑦)
67	9, 37	cfv 6349	. . . . . . . . . . . 12 class (comp‘𝑠)
68	65, 66, 67	co 7150	. . . . . . . . . . 11 class (⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))
69	61, 62, 68	co 7150	. . . . . . . . . 10 class ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))
70	60, 69	cop 4566	. . . . . . . . 9 class ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩
71	42, 43, 47, 48, 70	cmpo 7152	. . . . . . . 8 class (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)
72	39, 40, 41, 15, 71	cmpo 7152	. . . . . . 7 class (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))
73	38, 72	cop 4566	. . . . . 6 class ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩
74	33, 36, 73	ctp 4564	. . . . 5 class {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩}
75	12, 30, 74	csb 3882	. . . 4 class ⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩}
76	5, 11, 75	csb 3882	. . 3 class ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩}
77	2, 3, 4, 4, 76	cmpo 7152	. 2 class (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})
78	1, 77	wceq 1533	1 wff ×_c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})

Colors of variables: wff setvar class

This definition is referenced by: fnxpc 17420 xpcval 17421

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