Definition df-uncf 18182

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 18178	. 2 class uncurryF
2		vc	. . 3 setvar 𝑐
3		vf	. . 3 setvar 𝑓
4		cvv 3455	. . 3 class V
5		c1 11087	. . . . . 6 class 1
6	2	cv 1539	. . . . . 6 class 𝑐
7	5, 6	cfv 6519	. . . . 5 class (𝑐‘1)
8		c2 12252	. . . . . 6 class 2
9	8, 6	cfv 6519	. . . . 5 class (𝑐‘2)
10		cevlf 18176	. . . . 5 class evalF
11	7, 9, 10	co 7394	. . . 4 class ((𝑐‘1) evalF (𝑐‘2))
12	3	cv 1539	. . . . . 6 class 𝑓
13		cc0 11086	. . . . . . . 8 class 0
14	13, 6	cfv 6519	. . . . . . 7 class (𝑐‘0)
15		c1stf 18136	. . . . . . 7 class 1stF
16	14, 7, 15	co 7394	. . . . . 6 class ((𝑐‘0) 1stF (𝑐‘1))
17		ccofu 17824	. . . . . 6 class ∘func
18	12, 16, 17	co 7394	. . . . 5 class (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1)))
19		c2ndf 18137	. . . . . 6 class 2ndF
20	14, 7, 19	co 7394	. . . . 5 class ((𝑐‘0) 2ndF (𝑐‘1))
21		cprf 18138	. . . . 5 class ⟨,⟩F
22	18, 20, 21	co 7394	. . . 4 class ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))
23	11, 22, 17	co 7394	. . 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 7396	. 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 1540	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  18201