Definition df-uncf 18121

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 18117	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3436	. . 3 class V
5		c1 11007	. . . . . 6 class 1
6	2	cv 1540	. . . . . 6 class 𝑐
7	5, 6	cfv 6481	. . . . 5 class (𝑐‘1)
8		c2 12180	. . . . . 6 class 2
9	8, 6	cfv 6481	. . . . 5 class (𝑐‘2)
10		cevlf 18115	. . . . 5 class evalF
11	7, 9, 10	co 7346	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1540	. . . . . 6 class 𝑓
13		cc0 11006	. . . . . . . 8 class 0
14	13, 6	cfv 6481	. . . . . . 7 class (𝑐‘0)
15		c1stf 18075	. . . . . . 7 class 1stF
16	14, 7, 15	co 7346	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17763	. . . . . 6 class ∘func
18	12, 16, 17	co 7346	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18076	. . . . . 6 class 2ndF
20	14, 7, 19	co 7346	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18077	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7346	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7346	. . 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 7348	. 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 1541	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  18140