Definition df-uncf 18395

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 18391	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3450	. . 3 class V
5		c1 11205	. . . . . 6 class 1
6	2	cv 1568	. . . . . 6 class 𝑐
7	5, 6	cfv 6539	. . . . 5 class (𝑐‘1)
8		c2 12401	. . . . . 6 class 2
9	8, 6	cfv 6539	. . . . 5 class (𝑐‘2)
10		cevlf 18389	. . . . 5 class evalF
11	7, 9, 10	co 7421	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1568	. . . . . 6 class 𝑓
13		cc0 11204	. . . . . . . 8 class 0
14	13, 6	cfv 6539	. . . . . . 7 class (𝑐‘0)
15		c1stf 18349	. . . . . . 7 class 1stF
16	14, 7, 15	co 7421	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 18037	. . . . . 6 class ∘func
18	12, 16, 17	co 7421	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18350	. . . . . 6 class 2ndF
20	14, 7, 19	co 7421	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18351	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7421	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7421	. . 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 7423	. 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 1569	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 used by:  uncfval  18414