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Definition df-uncf 18109
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 18105 . 2 class uncurryF
2 vc . . 3 setvar 𝑐
3 vf . . 3 setvar 𝑓
4 cvv 3444 . . 3 class V
5 c1 11057 . . . . . 6 class 1
62cv 1541 . . . . . 6 class 𝑐
75, 6cfv 6497 . . . . 5 class (𝑐‘1)
8 c2 12213 . . . . . 6 class 2
98, 6cfv 6497 . . . . 5 class (𝑐‘2)
10 cevlf 18103 . . . . 5 class evalF
117, 9, 10co 7358 . . . 4 class ((𝑐‘1) evalF (𝑐‘2))
123cv 1541 . . . . . 6 class 𝑓
13 cc0 11056 . . . . . . . 8 class 0
1413, 6cfv 6497 . . . . . . 7 class (𝑐‘0)
15 c1stf 18062 . . . . . . 7 class 1stF
1614, 7, 15co 7358 . . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17 ccofu 17747 . . . . . 6 class func
1812, 16, 17co 7358 . . . . 5 class (𝑓func ((𝑐‘0) 1stF (𝑐‘1)))
19 c2ndf 18063 . . . . . 6 class 2ndF
2014, 7, 19co 7358 . . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21 cprf 18064 . . . . 5 class ⟨,⟩F
2218, 20, 21co 7358 . . . 4 class ((𝑓func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
2311, 22, 17co 7358 . . 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 7360 . 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 1542 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  18128
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