Definition df-uncf 18237

Description: Define the uncurry functor, which can be defined equationally using eval_F. 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 ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))))

Distinct variable group:   𝑓,𝑐

Detailed syntax breakdown of Definition df-uncf

Step	Hyp	Ref	Expression
1		cuncf 18233	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3453	. . 3 class V
5		c1 11067	. . . . . 6 class 1
6	2	cv 1558	. . . . . 6 class 𝑐
7	5, 6	cfv 6515	. . . . 5 class (𝑐‘1)
8		c2 12265	. . . . . 6 class 2
9	8, 6	cfv 6515	. . . . 5 class (𝑐‘2)
10		cevlf 18231	. . . . 5 class evalF
11	7, 9, 10	co 7390	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1558	. . . . . 6 class 𝑓
13		cc0 11066	. . . . . . . 8 class 0
14	13, 6	cfv 6515	. . . . . . 7 class (𝑐‘0)
15		c1stf 18191	. . . . . . 7 class 1stF
16	14, 7, 15	co 7390	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17879	. . . . . 6 class ∘func
18	12, 16, 17	co 7390	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18192	. . . . . 6 class 2ndF
20	14, 7, 19	co 7390	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18193	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7390	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7390	. . 3 class (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1))))
24	2, 3, 4, 4, 23	cmpo 7392	. 2 class (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))))
25	1, 24	wceq 1559	1 wff uncurryF = (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))))

This definition is referenced by:  uncfval  18256