Definition df-uncf 18285

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 18281	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3488	. . 3 class V
5		c1 11185	. . . . . 6 class 1
6	2	cv 1536	. . . . . 6 class 𝑐
7	5, 6	cfv 6573	. . . . 5 class (𝑐‘1)
8		c2 12348	. . . . . 6 class 2
9	8, 6	cfv 6573	. . . . 5 class (𝑐‘2)
10		cevlf 18279	. . . . 5 class evalF
11	7, 9, 10	co 7448	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1536	. . . . . 6 class 𝑓
13		cc0 11184	. . . . . . . 8 class 0
14	13, 6	cfv 6573	. . . . . . 7 class (𝑐‘0)
15		c1stf 18238	. . . . . . 7 class 1stF
16	14, 7, 15	co 7448	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17920	. . . . . 6 class ∘func
18	12, 16, 17	co 7448	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18239	. . . . . 6 class 2ndF
20	14, 7, 19	co 7448	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18240	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7448	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7448	. . 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 7450	. 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 1537	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  18304