Definition df-uncf 17677

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 17673	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3398	. . 3 class V
5		c1 10695	. . . . . 6 class 1
6	2	cv 1542	. . . . . 6 class 𝑐
7	5, 6	cfv 6358	. . . . 5 class (𝑐‘1)
8		c2 11850	. . . . . 6 class 2
9	8, 6	cfv 6358	. . . . 5 class (𝑐‘2)
10		cevlf 17671	. . . . 5 class evalF
11	7, 9, 10	co 7191	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1542	. . . . . 6 class 𝑓
13		cc0 10694	. . . . . . . 8 class 0
14	13, 6	cfv 6358	. . . . . . 7 class (𝑐‘0)
15		c1stf 17630	. . . . . . 7 class 1stF
16	14, 7, 15	co 7191	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17316	. . . . . 6 class ∘func
18	12, 16, 17	co 7191	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 17631	. . . . . 6 class 2ndF
20	14, 7, 19	co 7191	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 17632	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7191	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7191	. . 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 7193	. 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 1543	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  17696