Definition df-uncf 18382

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 18378	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3451	. . 3 class V
5		c1 11194	. . . . . 6 class 1
6	2	cv 1569	. . . . . 6 class 𝑐
7	5, 6	cfv 6537	. . . . 5 class (𝑐‘1)
8		c2 12390	. . . . . 6 class 2
9	8, 6	cfv 6537	. . . . 5 class (𝑐‘2)
10		cevlf 18376	. . . . 5 class evalF
11	7, 9, 10	co 7418	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1569	. . . . . 6 class 𝑓
13		cc0 11193	. . . . . . . 8 class 0
14	13, 6	cfv 6537	. . . . . . 7 class (𝑐‘0)
15		c1stf 18336	. . . . . . 7 class 1stF
16	14, 7, 15	co 7418	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 18024	. . . . . 6 class ∘func
18	12, 16, 17	co 7418	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18337	. . . . . 6 class 2ndF
20	14, 7, 19	co 7418	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18338	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7418	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7418	. . 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 7420	. 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  18401