Definition df-uncf 18303

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 18299	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3450	. . 3 class V
5		c1 11125	. . . . . 6 class 1
6	2	cv 1569	. . . . . 6 class 𝑐
7	5, 6	cfv 6533	. . . . 5 class (𝑐‘1)
8		c2 12319	. . . . . 6 class 2
9	8, 6	cfv 6533	. . . . 5 class (𝑐‘2)
10		cevlf 18297	. . . . 5 class evalF
11	7, 9, 10	co 7413	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1569	. . . . . 6 class 𝑓
13		cc0 11124	. . . . . . . 8 class 0
14	13, 6	cfv 6533	. . . . . . 7 class (𝑐‘0)
15		c1stf 18257	. . . . . . 7 class 1stF
16	14, 7, 15	co 7413	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17945	. . . . . 6 class ∘func
18	12, 16, 17	co 7413	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18258	. . . . . 6 class 2ndF
20	14, 7, 19	co 7413	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18259	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7413	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7413	. . 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 7415	. 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 1570	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  18322