Definition df-uncf 18172

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 18168	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3466	. . 3 class V
5		c1 11108	. . . . . 6 class 1
6	2	cv 1532	. . . . . 6 class 𝑐
7	5, 6	cfv 6534	. . . . 5 class (𝑐‘1)
8		c2 12265	. . . . . 6 class 2
9	8, 6	cfv 6534	. . . . 5 class (𝑐‘2)
10		cevlf 18166	. . . . 5 class evalF
11	7, 9, 10	co 7402	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1532	. . . . . 6 class 𝑓
13		cc0 11107	. . . . . . . 8 class 0
14	13, 6	cfv 6534	. . . . . . 7 class (𝑐‘0)
15		c1stf 18125	. . . . . . 7 class 1stF
16	14, 7, 15	co 7402	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17807	. . . . . 6 class ∘func
18	12, 16, 17	co 7402	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18126	. . . . . 6 class 2ndF
20	14, 7, 19	co 7402	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18127	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7402	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7402	. . 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 7404	. 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 1533	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  18191