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Definition df-xpc 17488

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 17484	. 2 class ×_c
2		vr	. . 3 setvar 𝑟
3		vs	. . 3 setvar 𝑠
4		cvv 3409	. . 3 class V
5		vb	. . . 4 setvar 𝑏
6	2	cv 1537	. . . . . 6 class 𝑟
7		cbs 16541	. . . . . 6 class Base
8	6, 7	cfv 6335	. . . . 5 class (Base‘𝑟)
9	3	cv 1537	. . . . . 6 class 𝑠
10	9, 7	cfv 6335	. . . . 5 class (Base‘𝑠)
11	8, 10	cxp 5522	. . . 4 class ((Base‘𝑟) × (Base‘𝑠))
12		vh	. . . . 5 setvar ℎ
13		vu	. . . . . 6 setvar 𝑢
14		vv	. . . . . 6 setvar 𝑣
15	5	cv 1537	. . . . . 6 class 𝑏
16	13	cv 1537	. . . . . . . . 9 class 𝑢
17		c1st 7691	. . . . . . . . 9 class 1^st
18	16, 17	cfv 6335	. . . . . . . 8 class (1^st ‘𝑢)
19	14	cv 1537	. . . . . . . . 9 class 𝑣
20	19, 17	cfv 6335	. . . . . . . 8 class (1^st ‘𝑣)
21		chom 16634	. . . . . . . . 9 class Hom
22	6, 21	cfv 6335	. . . . . . . 8 class (Hom ‘𝑟)
23	18, 20, 22	co 7150	. . . . . . 7 class ((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣))
24		c2nd 7692	. . . . . . . . 9 class 2^nd
25	16, 24	cfv 6335	. . . . . . . 8 class (2^nd ‘𝑢)
26	19, 24	cfv 6335	. . . . . . . 8 class (2^nd ‘𝑣)
27	9, 21	cfv 6335	. . . . . . . 8 class (Hom ‘𝑠)
28	25, 26, 27	co 7150	. . . . . . 7 class ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣))
29	23, 28	cxp 5522	. . . . . 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 16538	. . . . . . . 8 class ndx
32	31, 7	cfv 6335	. . . . . . 7 class (Base‘ndx)
33	32, 15	cop 4528	. . . . . 6 class ⟨(Base‘ndx), 𝑏⟩
34	31, 21	cfv 6335	. . . . . . 7 class (Hom ‘ndx)
35	12	cv 1537	. . . . . . 7 class ℎ
36	34, 35	cop 4528	. . . . . 6 class ⟨(Hom ‘ndx), ℎ⟩
37		cco 16635	. . . . . . . 8 class comp
38	31, 37	cfv 6335	. . . . . . 7 class (comp‘ndx)
39		vx	. . . . . . . 8 setvar 𝑥
40		vy	. . . . . . . 8 setvar 𝑦
41	15, 15	cxp 5522	. . . . . . . 8 class (𝑏 × 𝑏)
42		vg	. . . . . . . . 9 setvar 𝑔
43		vf	. . . . . . . . 9 setvar 𝑓
44	39	cv 1537	. . . . . . . . . . 11 class 𝑥
45	44, 24	cfv 6335	. . . . . . . . . 10 class (2^nd ‘𝑥)
46	40	cv 1537	. . . . . . . . . 10 class 𝑦
47	45, 46, 35	co 7150	. . . . . . . . 9 class ((2^nd ‘𝑥)ℎ𝑦)
48	44, 35	cfv 6335	. . . . . . . . 9 class (ℎ‘𝑥)
49	42	cv 1537	. . . . . . . . . . . 12 class 𝑔
50	49, 17	cfv 6335	. . . . . . . . . . 11 class (1^st ‘𝑔)
51	43	cv 1537	. . . . . . . . . . . 12 class 𝑓
52	51, 17	cfv 6335	. . . . . . . . . . 11 class (1^st ‘𝑓)
53	44, 17	cfv 6335	. . . . . . . . . . . . . 14 class (1^st ‘𝑥)
54	53, 17	cfv 6335	. . . . . . . . . . . . 13 class (1^st ‘(1^st ‘𝑥))
55	45, 17	cfv 6335	. . . . . . . . . . . . 13 class (1^st ‘(2^nd ‘𝑥))
56	54, 55	cop 4528	. . . . . . . . . . . 12 class ⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩
57	46, 17	cfv 6335	. . . . . . . . . . . 12 class (1^st ‘𝑦)
58	6, 37	cfv 6335	. . . . . . . . . . . 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 6335	. . . . . . . . . . 11 class (2^nd ‘𝑔)
62	51, 24	cfv 6335	. . . . . . . . . . 11 class (2^nd ‘𝑓)
63	53, 24	cfv 6335	. . . . . . . . . . . . 13 class (2^nd ‘(1^st ‘𝑥))
64	45, 24	cfv 6335	. . . . . . . . . . . . 13 class (2^nd ‘(2^nd ‘𝑥))
65	63, 64	cop 4528	. . . . . . . . . . . 12 class ⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩
66	46, 24	cfv 6335	. . . . . . . . . . . 12 class (2^nd ‘𝑦)
67	9, 37	cfv 6335	. . . . . . . . . . . 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 4528	. . . . . . . . 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 4528	. . . . . 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 4526	. . . . 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 3805	. . . 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 3805	. . 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 1538	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 17492 xpcval 17493

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