Definition df-uncf 18179

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 18175	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3432	. . 3 class V
5		c1 11037	. . . . . 6 class 1
6	2	cv 1546	. . . . . 6 class 𝑐
7	5, 6	cfv 6492	. . . . 5 class (𝑐‘1)
8		c2 12234	. . . . . 6 class 2
9	8, 6	cfv 6492	. . . . 5 class (𝑐‘2)
10		cevlf 18173	. . . . 5 class evalF
11	7, 9, 10	co 7363	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1546	. . . . . 6 class 𝑓
13		cc0 11036	. . . . . . . 8 class 0
14	13, 6	cfv 6492	. . . . . . 7 class (𝑐‘0)
15		c1stf 18133	. . . . . . 7 class 1stF
16	14, 7, 15	co 7363	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17821	. . . . . 6 class ∘func
18	12, 16, 17	co 7363	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18134	. . . . . 6 class 2ndF
20	14, 7, 19	co 7363	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18135	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7363	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7363	. . 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 7365	. 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 1547	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  18198