Definition df-uncf 17849

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 17845	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3422	. . 3 class V
5		c1 10803	. . . . . 6 class 1
6	2	cv 1538	. . . . . 6 class 𝑐
7	5, 6	cfv 6418	. . . . 5 class (𝑐‘1)
8		c2 11958	. . . . . 6 class 2
9	8, 6	cfv 6418	. . . . 5 class (𝑐‘2)
10		cevlf 17843	. . . . 5 class evalF
11	7, 9, 10	co 7255	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1538	. . . . . 6 class 𝑓
13		cc0 10802	. . . . . . . 8 class 0
14	13, 6	cfv 6418	. . . . . . 7 class (𝑐‘0)
15		c1stf 17802	. . . . . . 7 class 1stF
16	14, 7, 15	co 7255	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17487	. . . . . 6 class ∘func
18	12, 16, 17	co 7255	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 17803	. . . . . 6 class 2ndF
20	14, 7, 19	co 7255	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 17804	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7255	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7255	. . 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 7257	. 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 1539	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  17868