Definition df-uncf 18259

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 18255	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3457	. . 3 class V
5		c1 11089	. . . . . 6 class 1
6	2	cv 1562	. . . . . 6 class 𝑐
7	5, 6	cfv 6525	. . . . 5 class (𝑐‘1)
8		c2 12283	. . . . . 6 class 2
9	8, 6	cfv 6525	. . . . 5 class (𝑐‘2)
10		cevlf 18253	. . . . 5 class evalF
11	7, 9, 10	co 7400	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1562	. . . . . 6 class 𝑓
13		cc0 11088	. . . . . . . 8 class 0
14	13, 6	cfv 6525	. . . . . . 7 class (𝑐‘0)
15		c1stf 18213	. . . . . . 7 class 1stF
16	14, 7, 15	co 7400	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17901	. . . . . 6 class ∘func
18	12, 16, 17	co 7400	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18214	. . . . . 6 class 2ndF
20	14, 7, 19	co 7400	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18215	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7400	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7400	. . 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 7402	. 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 1563	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  18278