Definition df-uncf 18261

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 18257	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3479	. . 3 class V
5		c1 11157	. . . . . 6 class 1
6	2	cv 1538	. . . . . 6 class 𝑐
7	5, 6	cfv 6560	. . . . 5 class (𝑐‘1)
8		c2 12322	. . . . . 6 class 2
9	8, 6	cfv 6560	. . . . 5 class (𝑐‘2)
10		cevlf 18255	. . . . 5 class evalF
11	7, 9, 10	co 7432	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1538	. . . . . 6 class 𝑓
13		cc0 11156	. . . . . . . 8 class 0
14	13, 6	cfv 6560	. . . . . . 7 class (𝑐‘0)
15		c1stf 18215	. . . . . . 7 class 1stF
16	14, 7, 15	co 7432	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17902	. . . . . 6 class ∘func
18	12, 16, 17	co 7432	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18216	. . . . . 6 class 2ndF
20	14, 7, 19	co 7432	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18217	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7432	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7432	. . 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 7434	. 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  18280