Definition df-uncf 17933

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 17929	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3432	. . 3 class V
5		c1 10872	. . . . . 6 class 1
6	2	cv 1538	. . . . . 6 class 𝑐
7	5, 6	cfv 6433	. . . . 5 class (𝑐‘1)
8		c2 12028	. . . . . 6 class 2
9	8, 6	cfv 6433	. . . . 5 class (𝑐‘2)
10		cevlf 17927	. . . . 5 class evalF
11	7, 9, 10	co 7275	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1538	. . . . . 6 class 𝑓
13		cc0 10871	. . . . . . . 8 class 0
14	13, 6	cfv 6433	. . . . . . 7 class (𝑐‘0)
15		c1stf 17886	. . . . . . 7 class 1stF
16	14, 7, 15	co 7275	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17571	. . . . . 6 class ∘func
18	12, 16, 17	co 7275	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 17887	. . . . . 6 class 2ndF
20	14, 7, 19	co 7275	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 17888	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7275	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7275	. . 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 7277	. 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 1539	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  17952