Definition df-uncf 18176

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 18172	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3447	. . 3 class V
5		c1 11069	. . . . . 6 class 1
6	2	cv 1539	. . . . . 6 class 𝑐
7	5, 6	cfv 6511	. . . . 5 class (𝑐‘1)
8		c2 12241	. . . . . 6 class 2
9	8, 6	cfv 6511	. . . . 5 class (𝑐‘2)
10		cevlf 18170	. . . . 5 class evalF
11	7, 9, 10	co 7387	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1539	. . . . . 6 class 𝑓
13		cc0 11068	. . . . . . . 8 class 0
14	13, 6	cfv 6511	. . . . . . 7 class (𝑐‘0)
15		c1stf 18130	. . . . . . 7 class 1stF
16	14, 7, 15	co 7387	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17818	. . . . . 6 class ∘func
18	12, 16, 17	co 7387	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18131	. . . . . 6 class 2ndF
20	14, 7, 19	co 7387	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18132	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7387	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7387	. . 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 7389	. 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 1540	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  18195