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Definition df-xpc 17889

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 17885	. 2 class ×_c
2		vr	. . 3 setvar 𝑟
3		vs	. . 3 setvar 𝑠
4		cvv 3432	. . 3 class V
5		vb	. . . 4 setvar 𝑏
6	2	cv 1538	. . . . . 6 class 𝑟
7		cbs 16912	. . . . . 6 class Base
8	6, 7	cfv 6433	. . . . 5 class (Base‘𝑟)
9	3	cv 1538	. . . . . 6 class 𝑠
10	9, 7	cfv 6433	. . . . 5 class (Base‘𝑠)
11	8, 10	cxp 5587	. . . 4 class ((Base‘𝑟) × (Base‘𝑠))
12		vh	. . . . 5 setvar ℎ
13		vu	. . . . . 6 setvar 𝑢
14		vv	. . . . . 6 setvar 𝑣
15	5	cv 1538	. . . . . 6 class 𝑏
16	13	cv 1538	. . . . . . . . 9 class 𝑢
17		c1st 7829	. . . . . . . . 9 class 1^st
18	16, 17	cfv 6433	. . . . . . . 8 class (1^st ‘𝑢)
19	14	cv 1538	. . . . . . . . 9 class 𝑣
20	19, 17	cfv 6433	. . . . . . . 8 class (1^st ‘𝑣)
21		chom 16973	. . . . . . . . 9 class Hom
22	6, 21	cfv 6433	. . . . . . . 8 class (Hom ‘𝑟)
23	18, 20, 22	co 7275	. . . . . . 7 class ((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣))
24		c2nd 7830	. . . . . . . . 9 class 2^nd
25	16, 24	cfv 6433	. . . . . . . 8 class (2^nd ‘𝑢)
26	19, 24	cfv 6433	. . . . . . . 8 class (2^nd ‘𝑣)
27	9, 21	cfv 6433	. . . . . . . 8 class (Hom ‘𝑠)
28	25, 26, 27	co 7275	. . . . . . 7 class ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣))
29	23, 28	cxp 5587	. . . . . 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 7277	. . . . 5 class (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣))))
31		cnx 16894	. . . . . . . 8 class ndx
32	31, 7	cfv 6433	. . . . . . 7 class (Base‘ndx)
33	32, 15	cop 4567	. . . . . 6 class ⟨(Base‘ndx), 𝑏⟩
34	31, 21	cfv 6433	. . . . . . 7 class (Hom ‘ndx)
35	12	cv 1538	. . . . . . 7 class ℎ
36	34, 35	cop 4567	. . . . . 6 class ⟨(Hom ‘ndx), ℎ⟩
37		cco 16974	. . . . . . . 8 class comp
38	31, 37	cfv 6433	. . . . . . 7 class (comp‘ndx)
39		vx	. . . . . . . 8 setvar 𝑥
40		vy	. . . . . . . 8 setvar 𝑦
41	15, 15	cxp 5587	. . . . . . . 8 class (𝑏 × 𝑏)
42		vg	. . . . . . . . 9 setvar 𝑔
43		vf	. . . . . . . . 9 setvar 𝑓
44	39	cv 1538	. . . . . . . . . . 11 class 𝑥
45	44, 24	cfv 6433	. . . . . . . . . 10 class (2^nd ‘𝑥)
46	40	cv 1538	. . . . . . . . . 10 class 𝑦
47	45, 46, 35	co 7275	. . . . . . . . 9 class ((2^nd ‘𝑥)ℎ𝑦)
48	44, 35	cfv 6433	. . . . . . . . 9 class (ℎ‘𝑥)
49	42	cv 1538	. . . . . . . . . . . 12 class 𝑔
50	49, 17	cfv 6433	. . . . . . . . . . 11 class (1^st ‘𝑔)
51	43	cv 1538	. . . . . . . . . . . 12 class 𝑓
52	51, 17	cfv 6433	. . . . . . . . . . 11 class (1^st ‘𝑓)
53	44, 17	cfv 6433	. . . . . . . . . . . . . 14 class (1^st ‘𝑥)
54	53, 17	cfv 6433	. . . . . . . . . . . . 13 class (1^st ‘(1^st ‘𝑥))
55	45, 17	cfv 6433	. . . . . . . . . . . . 13 class (1^st ‘(2^nd ‘𝑥))
56	54, 55	cop 4567	. . . . . . . . . . . 12 class ⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩
57	46, 17	cfv 6433	. . . . . . . . . . . 12 class (1^st ‘𝑦)
58	6, 37	cfv 6433	. . . . . . . . . . . 12 class (comp‘𝑟)
59	56, 57, 58	co 7275	. . . . . . . . . . 11 class (⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))
60	50, 52, 59	co 7275	. . . . . . . . . 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 6433	. . . . . . . . . . 11 class (2^nd ‘𝑔)
62	51, 24	cfv 6433	. . . . . . . . . . 11 class (2^nd ‘𝑓)
63	53, 24	cfv 6433	. . . . . . . . . . . . 13 class (2^nd ‘(1^st ‘𝑥))
64	45, 24	cfv 6433	. . . . . . . . . . . . 13 class (2^nd ‘(2^nd ‘𝑥))
65	63, 64	cop 4567	. . . . . . . . . . . 12 class ⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩
66	46, 24	cfv 6433	. . . . . . . . . . . 12 class (2^nd ‘𝑦)
67	9, 37	cfv 6433	. . . . . . . . . . . 12 class (comp‘𝑠)
68	65, 66, 67	co 7275	. . . . . . . . . . 11 class (⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))
69	61, 62, 68	co 7275	. . . . . . . . . 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 4567	. . . . . . . . 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 7277	. . . . . . . 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 7277	. . . . . . 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 4567	. . . . . 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 4565	. . . . 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 3832	. . . 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 3832	. . 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 7277	. 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 1539	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 17893 xpcval 17894

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