Definition df-uncf 18181

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 18177	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3429	. . 3 class V
5		c1 11039	. . . . . 6 class 1
6	2	cv 1541	. . . . . 6 class 𝑐
7	5, 6	cfv 6498	. . . . 5 class (𝑐‘1)
8		c2 12236	. . . . . 6 class 2
9	8, 6	cfv 6498	. . . . 5 class (𝑐‘2)
10		cevlf 18175	. . . . 5 class evalF
11	7, 9, 10	co 7367	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1541	. . . . . 6 class 𝑓
13		cc0 11038	. . . . . . . 8 class 0
14	13, 6	cfv 6498	. . . . . . 7 class (𝑐‘0)
15		c1stf 18135	. . . . . . 7 class 1stF
16	14, 7, 15	co 7367	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17823	. . . . . 6 class ∘func
18	12, 16, 17	co 7367	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18136	. . . . . 6 class 2ndF
20	14, 7, 19	co 7367	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18137	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7367	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7367	. . 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 7369	. 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  18200