Definition df-uncf 18439

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 18435	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3454	. . 3 class V
5		c1 11240	. . . . . 6 class 1
6	2	cv 1569	. . . . . 6 class 𝑐
7	5, 6	cfv 6566	. . . . 5 class (𝑐‘1)
8		c2 12437	. . . . . 6 class 2
9	8, 6	cfv 6566	. . . . 5 class (𝑐‘2)
10		cevlf 18433	. . . . 5 class evalF
11	7, 9, 10	co 7451	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1569	. . . . . 6 class 𝑓
13		cc0 11239	. . . . . . . 8 class 0
14	13, 6	cfv 6566	. . . . . . 7 class (𝑐‘0)
15		c1stf 18393	. . . . . . 7 class 1stF
16	14, 7, 15	co 7451	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 18081	. . . . . 6 class ∘func
18	12, 16, 17	co 7451	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18394	. . . . . 6 class 2ndF
20	14, 7, 19	co 7451	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18395	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7451	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7451	. . 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 7453	. 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  18458