Definition df-uncf 17209

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 17205	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3415	. . 3 class V
5		c1 10254	. . . . . 6 class 1
6	2	cv 1657	. . . . . 6 class 𝑐
7	5, 6	cfv 6124	. . . . 5 class (𝑐‘1)
8		c2 11407	. . . . . 6 class 2
9	8, 6	cfv 6124	. . . . 5 class (𝑐‘2)
10		cevlf 17203	. . . . 5 class evalF
11	7, 9, 10	co 6906	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1657	. . . . . 6 class 𝑓
13		cc0 10253	. . . . . . . 8 class 0
14	13, 6	cfv 6124	. . . . . . 7 class (𝑐‘0)
15		c1stf 17163	. . . . . . 7 class 1stF
16	14, 7, 15	co 6906	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 16869	. . . . . 6 class ∘func
18	12, 16, 17	co 6906	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 17164	. . . . . 6 class 2ndF
20	14, 7, 19	co 6906	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 17165	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 6906	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 6906	. . 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	cmpt2 6908	. 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 1658	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  17228