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Definition df-uncf 17461
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 17457 . 2 class uncurryF
2 vc . . 3 setvar 𝑐
3 vf . . 3 setvar 𝑓
4 cvv 3444 . . 3 class V
5 c1 10531 . . . . . 6 class 1
62cv 1537 . . . . . 6 class 𝑐
75, 6cfv 6328 . . . . 5 class (𝑐‘1)
8 c2 11684 . . . . . 6 class 2
98, 6cfv 6328 . . . . 5 class (𝑐‘2)
10 cevlf 17455 . . . . 5 class evalF
117, 9, 10co 7139 . . . 4 class ((𝑐‘1) evalF (𝑐‘2))
123cv 1537 . . . . . 6 class 𝑓
13 cc0 10530 . . . . . . . 8 class 0
1413, 6cfv 6328 . . . . . . 7 class (𝑐‘0)
15 c1stf 17415 . . . . . . 7 class 1stF
1614, 7, 15co 7139 . . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17 ccofu 17122 . . . . . 6 class func
1812, 16, 17co 7139 . . . . 5 class (𝑓func ((𝑐‘0) 1stF (𝑐‘1)))
19 c2ndf 17416 . . . . . 6 class 2ndF
2014, 7, 19co 7139 . . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21 cprf 17417 . . . . 5 class ⟨,⟩F
2218, 20, 21co 7139 . . . 4 class ((𝑓func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
2311, 22, 17co 7139 . . 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 7141 . 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 1538 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  17480
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