Definition df-uncf 18167

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 18163	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3474	. . 3 class V
5		c1 11110	. . . . . 6 class 1
6	2	cv 1540	. . . . . 6 class 𝑐
7	5, 6	cfv 6543	. . . . 5 class (𝑐‘1)
8		c2 12266	. . . . . 6 class 2
9	8, 6	cfv 6543	. . . . 5 class (𝑐‘2)
10		cevlf 18161	. . . . 5 class evalF
11	7, 9, 10	co 7408	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1540	. . . . . 6 class 𝑓
13		cc0 11109	. . . . . . . 8 class 0
14	13, 6	cfv 6543	. . . . . . 7 class (𝑐‘0)
15		c1stf 18120	. . . . . . 7 class 1stF
16	14, 7, 15	co 7408	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17805	. . . . . 6 class ∘func
18	12, 16, 17	co 7408	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18121	. . . . . 6 class 2ndF
20	14, 7, 19	co 7408	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18122	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7408	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7408	. . 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 7410	. 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 1541	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  18186