Definition df-uncf 18138

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 18134	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3440	. . 3 class V
5		c1 11027	. . . . . 6 class 1
6	2	cv 1540	. . . . . 6 class 𝑐
7	5, 6	cfv 6492	. . . . 5 class (𝑐‘1)
8		c2 12200	. . . . . 6 class 2
9	8, 6	cfv 6492	. . . . 5 class (𝑐‘2)
10		cevlf 18132	. . . . 5 class evalF
11	7, 9, 10	co 7358	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1540	. . . . . 6 class 𝑓
13		cc0 11026	. . . . . . . 8 class 0
14	13, 6	cfv 6492	. . . . . . 7 class (𝑐‘0)
15		c1stf 18092	. . . . . . 7 class 1stF
16	14, 7, 15	co 7358	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17780	. . . . . 6 class ∘func
18	12, 16, 17	co 7358	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18093	. . . . . 6 class 2ndF
20	14, 7, 19	co 7358	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18094	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7358	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7358	. . 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 7360	. 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  18157