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Definition df-uncf 18172
Description: Define the uncurry functor, which can be defined equationally using evalF. Strictly speaking, the third category argument is not needed, since the resulting functor is extensionally equal regardless, but it is used in the equational definition and is too much work to remove. (Contributed by Mario Carneiro, 13-Jan-2017.)
Assertion
Ref Expression
df-uncf uncurryF = (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))))
Distinct variable group:   𝑓,𝑐

Detailed syntax breakdown of Definition df-uncf
StepHypRef Expression
1 cuncf 18168 . 2 class uncurryF
2 vc . . 3 setvar 𝑐
3 vf . . 3 setvar 𝑓
4 cvv 3466 . . 3 class V
5 c1 11108 . . . . . 6 class 1
62cv 1532 . . . . . 6 class 𝑐
75, 6cfv 6534 . . . . 5 class (𝑐‘1)
8 c2 12265 . . . . . 6 class 2
98, 6cfv 6534 . . . . 5 class (𝑐‘2)
10 cevlf 18166 . . . . 5 class evalF
117, 9, 10co 7402 . . . 4 class ((𝑐‘1) evalF (𝑐‘2))
123cv 1532 . . . . . 6 class 𝑓
13 cc0 11107 . . . . . . . 8 class 0
1413, 6cfv 6534 . . . . . . 7 class (𝑐‘0)
15 c1stf 18125 . . . . . . 7 class 1stF
1614, 7, 15co 7402 . . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17 ccofu 17807 . . . . . 6 class func
1812, 16, 17co 7402 . . . . 5 class (𝑓func ((𝑐‘0) 1stF (𝑐‘1)))
19 c2ndf 18126 . . . . . 6 class 2ndF
2014, 7, 19co 7402 . . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21 cprf 18127 . . . . 5 class ⟨,⟩F
2218, 20, 21co 7402 . . . 4 class ((𝑓func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
2311, 22, 17co 7402 . . 3 class (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1))))
242, 3, 4, 4, 23cmpo 7404 . 2 class (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))))
251, 24wceq 1533 1 wff uncurryF = (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))))
Colors of variables: wff setvar class
This definition is referenced by:  uncfval  18191
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