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Definition df-uncf 18272
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 18268 . 2 class uncurryF
2 vc . . 3 setvar 𝑐
3 vf . . 3 setvar 𝑓
4 cvv 3478 . . 3 class V
5 c1 11154 . . . . . 6 class 1
62cv 1536 . . . . . 6 class 𝑐
75, 6cfv 6563 . . . . 5 class (𝑐‘1)
8 c2 12319 . . . . . 6 class 2
98, 6cfv 6563 . . . . 5 class (𝑐‘2)
10 cevlf 18266 . . . . 5 class evalF
117, 9, 10co 7431 . . . 4 class ((𝑐‘1) evalF (𝑐‘2))
123cv 1536 . . . . . 6 class 𝑓
13 cc0 11153 . . . . . . . 8 class 0
1413, 6cfv 6563 . . . . . . 7 class (𝑐‘0)
15 c1stf 18225 . . . . . . 7 class 1stF
1614, 7, 15co 7431 . . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17 ccofu 17907 . . . . . 6 class func
1812, 16, 17co 7431 . . . . 5 class (𝑓func ((𝑐‘0) 1stF (𝑐‘1)))
19 c2ndf 18226 . . . . . 6 class 2ndF
2014, 7, 19co 7431 . . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21 cprf 18227 . . . . 5 class ⟨,⟩F
2218, 20, 21co 7431 . . . 4 class ((𝑓func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
2311, 22, 17co 7431 . . 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 7433 . 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 1537 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  18291
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