Definition df-uncf 18150

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 18146	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3442	. . 3 class V
5		c1 11039	. . . . . 6 class 1
6	2	cv 1541	. . . . . 6 class 𝑐
7	5, 6	cfv 6500	. . . . 5 class (𝑐‘1)
8		c2 12212	. . . . . 6 class 2
9	8, 6	cfv 6500	. . . . 5 class (𝑐‘2)
10		cevlf 18144	. . . . 5 class evalF
11	7, 9, 10	co 7368	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1541	. . . . . 6 class 𝑓
13		cc0 11038	. . . . . . . 8 class 0
14	13, 6	cfv 6500	. . . . . . 7 class (𝑐‘0)
15		c1stf 18104	. . . . . . 7 class 1stF
16	14, 7, 15	co 7368	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17792	. . . . . 6 class ∘func
18	12, 16, 17	co 7368	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18105	. . . . . 6 class 2ndF
20	14, 7, 19	co 7368	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18106	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7368	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7368	. . 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 7370	. 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 1542	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  18169