Definition df-uncf 18288

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 18284	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3457	. . 3 class V
5		c1 11112	. . . . . 6 class 1
6	2	cv 1569	. . . . . 6 class 𝑐
7	5, 6	cfv 6540	. . . . 5 class (𝑐‘1)
8		c2 12306	. . . . . 6 class 2
9	8, 6	cfv 6540	. . . . 5 class (𝑐‘2)
10		cevlf 18282	. . . . 5 class evalF
11	7, 9, 10	co 7416	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1569	. . . . . 6 class 𝑓
13		cc0 11111	. . . . . . . 8 class 0
14	13, 6	cfv 6540	. . . . . . 7 class (𝑐‘0)
15		c1stf 18242	. . . . . . 7 class 1stF
16	14, 7, 15	co 7416	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17930	. . . . . 6 class ∘func
18	12, 16, 17	co 7416	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18243	. . . . . 6 class 2ndF
20	14, 7, 19	co 7416	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18244	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7416	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7416	. . 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 7418	. 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 used by:  uncfval  18307