Definition df-uncf 18272

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 18268	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3455	. . 3 class V
5		c1 11102	. . . . . 6 class 1
6	2	cv 1569	. . . . . 6 class 𝑐
7	5, 6	cfv 6538	. . . . 5 class (𝑐‘1)
8		c2 12296	. . . . . 6 class 2
9	8, 6	cfv 6538	. . . . 5 class (𝑐‘2)
10		cevlf 18266	. . . . 5 class evalF
11	7, 9, 10	co 7412	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1569	. . . . . 6 class 𝑓
13		cc0 11101	. . . . . . . 8 class 0
14	13, 6	cfv 6538	. . . . . . 7 class (𝑐‘0)
15		c1stf 18226	. . . . . . 7 class 1stF
16	14, 7, 15	co 7412	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17914	. . . . . 6 class ∘func
18	12, 16, 17	co 7412	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18227	. . . . . 6 class 2ndF
20	14, 7, 19	co 7412	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18228	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7412	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7412	. . 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 7414	. 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 1570	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  18291