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Definition df-uncf 14005
Description: Define the uncurry functor, which can be defined equationally using evalF. 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  e. 
_V ,  f  e. 
_V  |->  ( ( ( c `  1 ) evalF  ( c `  2 ) )  o.func  ( ( f  o.func  ( ( c `  0
)  1stF  ( c `  1
) ) ) ⟨,⟩F  ( ( c ` 
0 )  2ndF  ( c `  1 ) ) ) ) )
Distinct variable group:    f, c

Detailed syntax breakdown of Definition df-uncf
StepHypRef Expression
1 cuncf 14001 . 2  class uncurryF
2 vc . . 3  set  c
3 vf . . 3  set  f
4 cvv 2801 . . 3  class  _V
5 c1 8754 . . . . . 6  class  1
62cv 1631 . . . . . 6  class  c
75, 6cfv 5271 . . . . 5  class  ( c `
 1 )
8 c2 9811 . . . . . 6  class  2
98, 6cfv 5271 . . . . 5  class  ( c `
 2 )
10 cevlf 13999 . . . . 5  class evalF
117, 9, 10co 5874 . . . 4  class  ( ( c `  1 ) evalF  ( c `  2 ) )
123cv 1631 . . . . . 6  class  f
13 cc0 8753 . . . . . . . 8  class  0
1413, 6cfv 5271 . . . . . . 7  class  ( c `
 0 )
15 c1stf 13959 . . . . . . 7  class  1stF
1614, 7, 15co 5874 . . . . . 6  class  ( ( c `  0 )  1stF  ( c `  1
) )
17 ccofu 13746 . . . . . 6  class  o.func
1812, 16, 17co 5874 . . . . 5  class  ( f  o.func  ( ( c ` 
0 )  1stF  ( c `  1 ) ) )
19 c2ndf 13960 . . . . . 6  class  2ndF
2014, 7, 19co 5874 . . . . 5  class  ( ( c `  0 )  2ndF  ( c `  1
) )
21 cprf 13961 . . . . 5  class ⟨,⟩F
2218, 20, 21co 5874 . . . 4  class  ( ( f  o.func  ( ( c ` 
0 )  1stF  ( c `  1 ) ) ) ⟨,⟩F  ( ( c ` 
0 )  2ndF  ( c `  1 ) ) )
2311, 22, 17co 5874 . . 3  class  ( ( ( c `  1
) evalF 
( c `  2
) )  o.func  ( (
f  o.func  ( ( c ` 
0 )  1stF  ( c `  1 ) ) ) ⟨,⟩F  ( ( c ` 
0 )  2ndF  ( c `  1 ) ) ) )
242, 3, 4, 4, 23cmpt2 5876 . 2  class  ( c  e.  _V ,  f  e.  _V  |->  ( ( ( c `  1
) evalF 
( c `  2
) )  o.func  ( (
f  o.func  ( ( c ` 
0 )  1stF  ( c `  1 ) ) ) ⟨,⟩F  ( ( c ` 
0 )  2ndF  ( c `  1 ) ) ) ) )
251, 24wceq 1632 1  wff uncurryF  =  ( c  e. 
_V ,  f  e. 
_V  |->  ( ( ( c `  1 ) evalF  ( c `  2 ) )  o.func  ( ( f  o.func  ( ( c `  0
)  1stF  ( c `  1
) ) ) ⟨,⟩F  ( ( c ` 
0 )  2ndF  ( c `  1 ) ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  uncfval  14024
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