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Definition df-uncf 15358
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 15354 . 2  class uncurryF
2 vc . . 3  setvar  c
3 vf . . 3  setvar  f
4 cvv 3095 . . 3  class  _V
5 c1 9496 . . . . . 6  class  1
62cv 1382 . . . . . 6  class  c
75, 6cfv 5578 . . . . 5  class  ( c `
 1 )
8 c2 10591 . . . . . 6  class  2
98, 6cfv 5578 . . . . 5  class  ( c `
 2 )
10 cevlf 15352 . . . . 5  class evalF
117, 9, 10co 6281 . . . 4  class  ( ( c `  1 ) evalF  ( c `  2 ) )
123cv 1382 . . . . . 6  class  f
13 cc0 9495 . . . . . . . 8  class  0
1413, 6cfv 5578 . . . . . . 7  class  ( c `
 0 )
15 c1stf 15312 . . . . . . 7  class  1stF
1614, 7, 15co 6281 . . . . . 6  class  ( ( c `  0 )  1stF  ( c `  1
) )
17 ccofu 15099 . . . . . 6  class  o.func
1812, 16, 17co 6281 . . . . 5  class  ( f  o.func  ( ( c ` 
0 )  1stF  ( c `  1 ) ) )
19 c2ndf 15313 . . . . . 6  class  2ndF
2014, 7, 19co 6281 . . . . 5  class  ( ( c `  0 )  2ndF  ( c `  1
) )
21 cprf 15314 . . . . 5  class ⟨,⟩F
2218, 20, 21co 6281 . . . 4  class  ( ( f  o.func  ( ( c ` 
0 )  1stF  ( c `  1 ) ) ) ⟨,⟩F  ( ( c ` 
0 )  2ndF  ( c `  1 ) ) )
2311, 22, 17co 6281 . . 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 6283 . 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 1383 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  15377
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