Definition df-uncf 16855

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 16851	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3200	. . 3 class V
5		c1 9937	. . . . . 6 class 1
6	2	cv 1482	. . . . . 6 class 𝑐
7	5, 6	cfv 5888	. . . . 5 class (𝑐‘1)
8		c2 11070	. . . . . 6 class 2
9	8, 6	cfv 5888	. . . . 5 class (𝑐‘2)
10		cevlf 16849	. . . . 5 class evalF
11	7, 9, 10	co 6650	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1482	. . . . . 6 class 𝑓
13		cc0 9936	. . . . . . . 8 class 0
14	13, 6	cfv 5888	. . . . . . 7 class (𝑐‘0)
15		c1stf 16809	. . . . . . 7 class 1stF
16	14, 7, 15	co 6650	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 16516	. . . . . 6 class ∘func
18	12, 16, 17	co 6650	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 16810	. . . . . 6 class 2ndF
20	14, 7, 19	co 6650	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 16811	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 6650	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 6650	. . 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 6652	. 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 1483	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  16874