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Definition df-xpc 18078

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 18074	. 2 class ×_c
2		vr	. . 3 setvar 𝑟
3		vs	. . 3 setvar 𝑠
4		cvv 3436	. . 3 class V
5		vb	. . . 4 setvar 𝑏
6	2	cv 1539	. . . . . 6 class 𝑟
7		cbs 17120	. . . . . 6 class Base
8	6, 7	cfv 6482	. . . . 5 class (Base‘𝑟)
9	3	cv 1539	. . . . . 6 class 𝑠
10	9, 7	cfv 6482	. . . . 5 class (Base‘𝑠)
11	8, 10	cxp 5617	. . . 4 class ((Base‘𝑟) × (Base‘𝑠))
12		vh	. . . . 5 setvar ℎ
13		vu	. . . . . 6 setvar 𝑢
14		vv	. . . . . 6 setvar 𝑣
15	5	cv 1539	. . . . . 6 class 𝑏
16	13	cv 1539	. . . . . . . . 9 class 𝑢
17		c1st 7922	. . . . . . . . 9 class 1^st
18	16, 17	cfv 6482	. . . . . . . 8 class (1^st ‘𝑢)
19	14	cv 1539	. . . . . . . . 9 class 𝑣
20	19, 17	cfv 6482	. . . . . . . 8 class (1^st ‘𝑣)
21		chom 17172	. . . . . . . . 9 class Hom
22	6, 21	cfv 6482	. . . . . . . 8 class (Hom ‘𝑟)
23	18, 20, 22	co 7349	. . . . . . 7 class ((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣))
24		c2nd 7923	. . . . . . . . 9 class 2^nd
25	16, 24	cfv 6482	. . . . . . . 8 class (2^nd ‘𝑢)
26	19, 24	cfv 6482	. . . . . . . 8 class (2^nd ‘𝑣)
27	9, 21	cfv 6482	. . . . . . . 8 class (Hom ‘𝑠)
28	25, 26, 27	co 7349	. . . . . . 7 class ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣))
29	23, 28	cxp 5617	. . . . . 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 7351	. . . . 5 class (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣))))
31		cnx 17104	. . . . . . . 8 class ndx
32	31, 7	cfv 6482	. . . . . . 7 class (Base‘ndx)
33	32, 15	cop 4583	. . . . . 6 class ⟨(Base‘ndx), 𝑏⟩
34	31, 21	cfv 6482	. . . . . . 7 class (Hom ‘ndx)
35	12	cv 1539	. . . . . . 7 class ℎ
36	34, 35	cop 4583	. . . . . 6 class ⟨(Hom ‘ndx), ℎ⟩
37		cco 17173	. . . . . . . 8 class comp
38	31, 37	cfv 6482	. . . . . . 7 class (comp‘ndx)
39		vx	. . . . . . . 8 setvar 𝑥
40		vy	. . . . . . . 8 setvar 𝑦
41	15, 15	cxp 5617	. . . . . . . 8 class (𝑏 × 𝑏)
42		vg	. . . . . . . . 9 setvar 𝑔
43		vf	. . . . . . . . 9 setvar 𝑓
44	39	cv 1539	. . . . . . . . . . 11 class 𝑥
45	44, 24	cfv 6482	. . . . . . . . . 10 class (2^nd ‘𝑥)
46	40	cv 1539	. . . . . . . . . 10 class 𝑦
47	45, 46, 35	co 7349	. . . . . . . . 9 class ((2^nd ‘𝑥)ℎ𝑦)
48	44, 35	cfv 6482	. . . . . . . . 9 class (ℎ‘𝑥)
49	42	cv 1539	. . . . . . . . . . . 12 class 𝑔
50	49, 17	cfv 6482	. . . . . . . . . . 11 class (1^st ‘𝑔)
51	43	cv 1539	. . . . . . . . . . . 12 class 𝑓
52	51, 17	cfv 6482	. . . . . . . . . . 11 class (1^st ‘𝑓)
53	44, 17	cfv 6482	. . . . . . . . . . . . . 14 class (1^st ‘𝑥)
54	53, 17	cfv 6482	. . . . . . . . . . . . 13 class (1^st ‘(1^st ‘𝑥))
55	45, 17	cfv 6482	. . . . . . . . . . . . 13 class (1^st ‘(2^nd ‘𝑥))
56	54, 55	cop 4583	. . . . . . . . . . . 12 class ⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩
57	46, 17	cfv 6482	. . . . . . . . . . . 12 class (1^st ‘𝑦)
58	6, 37	cfv 6482	. . . . . . . . . . . 12 class (comp‘𝑟)
59	56, 57, 58	co 7349	. . . . . . . . . . 11 class (⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))
60	50, 52, 59	co 7349	. . . . . . . . . 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 6482	. . . . . . . . . . 11 class (2^nd ‘𝑔)
62	51, 24	cfv 6482	. . . . . . . . . . 11 class (2^nd ‘𝑓)
63	53, 24	cfv 6482	. . . . . . . . . . . . 13 class (2^nd ‘(1^st ‘𝑥))
64	45, 24	cfv 6482	. . . . . . . . . . . . 13 class (2^nd ‘(2^nd ‘𝑥))
65	63, 64	cop 4583	. . . . . . . . . . . 12 class ⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩
66	46, 24	cfv 6482	. . . . . . . . . . . 12 class (2^nd ‘𝑦)
67	9, 37	cfv 6482	. . . . . . . . . . . 12 class (comp‘𝑠)
68	65, 66, 67	co 7349	. . . . . . . . . . 11 class (⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))
69	61, 62, 68	co 7349	. . . . . . . . . 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 4583	. . . . . . . . 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 7351	. . . . . . . 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 7351	. . . . . . 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 4583	. . . . . 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 4581	. . . . 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 3851	. . . 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 3851	. . 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 7351	. 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 1540	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 18082 xpcval 18083 reldmxpcALT 49232

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