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Definition df-xpc 17209
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(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩})
Distinct variable group:   𝑓,𝑏,𝑔,,𝑟,𝑠,𝑢,𝑣,𝑥,𝑦

Detailed syntax breakdown of Definition df-xpc
StepHypRef Expression
1 cxpc 17205 . 2 class ×c
2 vr . . 3 setvar 𝑟
3 vs . . 3 setvar 𝑠
4 cvv 3398 . . 3 class V
5 vb . . . 4 setvar 𝑏
62cv 1600 . . . . . 6 class 𝑟
7 cbs 16266 . . . . . 6 class Base
86, 7cfv 6137 . . . . 5 class (Base‘𝑟)
93cv 1600 . . . . . 6 class 𝑠
109, 7cfv 6137 . . . . 5 class (Base‘𝑠)
118, 10cxp 5355 . . . 4 class ((Base‘𝑟) × (Base‘𝑠))
12 vh . . . . 5 setvar
13 vu . . . . . 6 setvar 𝑢
14 vv . . . . . 6 setvar 𝑣
155cv 1600 . . . . . 6 class 𝑏
1613cv 1600 . . . . . . . . 9 class 𝑢
17 c1st 7445 . . . . . . . . 9 class 1st
1816, 17cfv 6137 . . . . . . . 8 class (1st𝑢)
1914cv 1600 . . . . . . . . 9 class 𝑣
2019, 17cfv 6137 . . . . . . . 8 class (1st𝑣)
21 chom 16360 . . . . . . . . 9 class Hom
226, 21cfv 6137 . . . . . . . 8 class (Hom ‘𝑟)
2318, 20, 22co 6924 . . . . . . 7 class ((1st𝑢)(Hom ‘𝑟)(1st𝑣))
24 c2nd 7446 . . . . . . . . 9 class 2nd
2516, 24cfv 6137 . . . . . . . 8 class (2nd𝑢)
2619, 24cfv 6137 . . . . . . . 8 class (2nd𝑣)
279, 21cfv 6137 . . . . . . . 8 class (Hom ‘𝑠)
2825, 26, 27co 6924 . . . . . . 7 class ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣))
2923, 28cxp 5355 . . . . . 6 class (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))
3013, 14, 15, 15, 29cmpt2 6926 . . . . 5 class (𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣))))
31 cnx 16263 . . . . . . . 8 class ndx
3231, 7cfv 6137 . . . . . . 7 class (Base‘ndx)
3332, 15cop 4404 . . . . . 6 class ⟨(Base‘ndx), 𝑏
3431, 21cfv 6137 . . . . . . 7 class (Hom ‘ndx)
3512cv 1600 . . . . . . 7 class
3634, 35cop 4404 . . . . . 6 class ⟨(Hom ‘ndx),
37 cco 16361 . . . . . . . 8 class comp
3831, 37cfv 6137 . . . . . . 7 class (comp‘ndx)
39 vx . . . . . . . 8 setvar 𝑥
40 vy . . . . . . . 8 setvar 𝑦
4115, 15cxp 5355 . . . . . . . 8 class (𝑏 × 𝑏)
42 vg . . . . . . . . 9 setvar 𝑔
43 vf . . . . . . . . 9 setvar 𝑓
4439cv 1600 . . . . . . . . . . 11 class 𝑥
4544, 24cfv 6137 . . . . . . . . . 10 class (2nd𝑥)
4640cv 1600 . . . . . . . . . 10 class 𝑦
4745, 46, 35co 6924 . . . . . . . . 9 class ((2nd𝑥)𝑦)
4844, 35cfv 6137 . . . . . . . . 9 class (𝑥)
4942cv 1600 . . . . . . . . . . . 12 class 𝑔
5049, 17cfv 6137 . . . . . . . . . . 11 class (1st𝑔)
5143cv 1600 . . . . . . . . . . . 12 class 𝑓
5251, 17cfv 6137 . . . . . . . . . . 11 class (1st𝑓)
5344, 17cfv 6137 . . . . . . . . . . . . . 14 class (1st𝑥)
5453, 17cfv 6137 . . . . . . . . . . . . 13 class (1st ‘(1st𝑥))
5545, 17cfv 6137 . . . . . . . . . . . . 13 class (1st ‘(2nd𝑥))
5654, 55cop 4404 . . . . . . . . . . . 12 class ⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩
5746, 17cfv 6137 . . . . . . . . . . . 12 class (1st𝑦)
586, 37cfv 6137 . . . . . . . . . . . 12 class (comp‘𝑟)
5956, 57, 58co 6924 . . . . . . . . . . 11 class (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))
6050, 52, 59co 6924 . . . . . . . . . 10 class ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓))
6149, 24cfv 6137 . . . . . . . . . . 11 class (2nd𝑔)
6251, 24cfv 6137 . . . . . . . . . . 11 class (2nd𝑓)
6353, 24cfv 6137 . . . . . . . . . . . . 13 class (2nd ‘(1st𝑥))
6445, 24cfv 6137 . . . . . . . . . . . . 13 class (2nd ‘(2nd𝑥))
6563, 64cop 4404 . . . . . . . . . . . 12 class ⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩
6646, 24cfv 6137 . . . . . . . . . . . 12 class (2nd𝑦)
679, 37cfv 6137 . . . . . . . . . . . 12 class (comp‘𝑠)
6865, 66, 67co 6924 . . . . . . . . . . 11 class (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))
6961, 62, 68co 6924 . . . . . . . . . 10 class ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))
7060, 69cop 4404 . . . . . . . . 9 class ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩
7142, 43, 47, 48, 70cmpt2 6926 . . . . . . . 8 class (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩)
7239, 40, 41, 15, 71cmpt2 6926 . . . . . . 7 class (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))
7338, 72cop 4404 . . . . . 6 class ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩
7433, 36, 73ctp 4402 . . . . 5 class {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩}
7512, 30, 74csb 3751 . . . 4 class (𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩}
765, 11, 75csb 3751 . . 3 class ((Base‘𝑟) × (Base‘𝑠)) / 𝑏(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩}
772, 3, 4, 4, 76cmpt2 6926 . 2 class (𝑟 ∈ V, 𝑠 ∈ V ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩})
781, 77wceq 1601 1 wff ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩})
Colors of variables: wff setvar class
This definition is referenced by:  fnxpc  17213  xpcval  17214
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