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Definition df-2ndf 16637
 Description: Define the second projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Assertion
Ref Expression
df-2ndf 2ndF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)
Distinct variable group:   𝑟,𝑏,𝑠,𝑥,𝑦

Detailed syntax breakdown of Definition df-2ndf
StepHypRef Expression
1 c2ndf 16633 . 2 class 2ndF
2 vr . . 3 setvar 𝑟
3 vs . . 3 setvar 𝑠
4 ccat 16148 . . 3 class Cat
5 vb . . . 4 setvar 𝑏
62cv 1474 . . . . . 6 class 𝑟
7 cbs 15695 . . . . . 6 class Base
86, 7cfv 5804 . . . . 5 class (Base‘𝑟)
93cv 1474 . . . . . 6 class 𝑠
109, 7cfv 5804 . . . . 5 class (Base‘𝑠)
118, 10cxp 5036 . . . 4 class ((Base‘𝑟) × (Base‘𝑠))
12 c2nd 7058 . . . . . 6 class 2nd
135cv 1474 . . . . . 6 class 𝑏
1412, 13cres 5040 . . . . 5 class (2nd𝑏)
15 vx . . . . . 6 setvar 𝑥
16 vy . . . . . 6 setvar 𝑦
1715cv 1474 . . . . . . . 8 class 𝑥
1816cv 1474 . . . . . . . 8 class 𝑦
19 cxpc 16631 . . . . . . . . . 10 class ×c
206, 9, 19co 6549 . . . . . . . . 9 class (𝑟 ×c 𝑠)
21 chom 15779 . . . . . . . . 9 class Hom
2220, 21cfv 5804 . . . . . . . 8 class (Hom ‘(𝑟 ×c 𝑠))
2317, 18, 22co 6549 . . . . . . 7 class (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)
2412, 23cres 5040 . . . . . 6 class (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦))
2515, 16, 13, 13, 24cmpt2 6551 . . . . 5 class (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))
2614, 25cop 4131 . . . 4 class ⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩
275, 11, 26csb 3499 . . 3 class ((Base‘𝑟) × (Base‘𝑠)) / 𝑏⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩
282, 3, 4, 4, 27cmpt2 6551 . 2 class (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)
291, 28wceq 1475 1 wff 2ndF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)
 Colors of variables: wff setvar class This definition is referenced by:  2ndfval  16657
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