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Definition df-uncf 15331
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 15327 . 2  class uncurryF
2 vc . . 3  setvar  c
3 vf . . 3  setvar  f
4 cvv 3106 . . 3  class  _V
5 c1 9482 . . . . . 6  class  1
62cv 1373 . . . . . 6  class  c
75, 6cfv 5579 . . . . 5  class  ( c `
 1 )
8 c2 10574 . . . . . 6  class  2
98, 6cfv 5579 . . . . 5  class  ( c `
 2 )
10 cevlf 15325 . . . . 5  class evalF
117, 9, 10co 6275 . . . 4  class  ( ( c `  1 ) evalF  ( c `  2 ) )
123cv 1373 . . . . . 6  class  f
13 cc0 9481 . . . . . . . 8  class  0
1413, 6cfv 5579 . . . . . . 7  class  ( c `
 0 )
15 c1stf 15285 . . . . . . 7  class  1stF
1614, 7, 15co 6275 . . . . . 6  class  ( ( c `  0 )  1stF  ( c `  1
) )
17 ccofu 15072 . . . . . 6  class  o.func
1812, 16, 17co 6275 . . . . 5  class  ( f  o.func  ( ( c ` 
0 )  1stF  ( c `  1 ) ) )
19 c2ndf 15286 . . . . . 6  class  2ndF
2014, 7, 19co 6275 . . . . 5  class  ( ( c `  0 )  2ndF  ( c `  1
) )
21 cprf 15287 . . . . 5  class ⟨,⟩F
2218, 20, 21co 6275 . . . 4  class  ( ( f  o.func  ( ( c ` 
0 )  1stF  ( c `  1 ) ) ) ⟨,⟩F  ( ( c ` 
0 )  2ndF  ( c `  1 ) ) )
2311, 22, 17co 6275 . . 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 6277 . 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 1374 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 setvar class
This definition is referenced by:  uncfval  15350
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