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Definition df-uncf 17453
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 17449 . 2 class uncurryF
2 vc . . 3 setvar 𝑐
3 vf . . 3 setvar 𝑓
4 cvv 3492 . . 3 class V
5 c1 10526 . . . . . 6 class 1
62cv 1527 . . . . . 6 class 𝑐
75, 6cfv 6348 . . . . 5 class (𝑐‘1)
8 c2 11680 . . . . . 6 class 2
98, 6cfv 6348 . . . . 5 class (𝑐‘2)
10 cevlf 17447 . . . . 5 class evalF
117, 9, 10co 7145 . . . 4 class ((𝑐‘1) evalF (𝑐‘2))
123cv 1527 . . . . . 6 class 𝑓
13 cc0 10525 . . . . . . . 8 class 0
1413, 6cfv 6348 . . . . . . 7 class (𝑐‘0)
15 c1stf 17407 . . . . . . 7 class 1stF
1614, 7, 15co 7145 . . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17 ccofu 17114 . . . . . 6 class func
1812, 16, 17co 7145 . . . . 5 class (𝑓func ((𝑐‘0) 1stF (𝑐‘1)))
19 c2ndf 17408 . . . . . 6 class 2ndF
2014, 7, 19co 7145 . . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21 cprf 17409 . . . . 5 class ⟨,⟩F
2218, 20, 21co 7145 . . . 4 class ((𝑓func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
2311, 22, 17co 7145 . . 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 7147 . 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 1528 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  17472
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