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Definition df-full 14796
Description: Function returning all the full functors from a category 
C to a category  D. A full functor is a functor in which all the morphism maps  G ( X ,  Y ) between objects  X ,  Y  e.  C are surjections. (Contributed by Mario Carneiro, 26-Jan-2017.)
Assertion
Ref Expression
df-full  |- Full  =  ( c  e.  Cat , 
d  e.  Cat  |->  {
<. f ,  g >.  |  ( f ( c  Func  d )
g  /\  A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) ( Hom  `  d ) ( f `
 y ) ) ) } )
Distinct variable group:    c, d, f, g, x, y

Detailed syntax breakdown of Definition df-full
StepHypRef Expression
1 cful 14794 . 2  class Full
2 vc . . 3  setvar  c
3 vd . . 3  setvar  d
4 ccat 14584 . . 3  class  Cat
5 vf . . . . . . 7  setvar  f
65cv 1361 . . . . . 6  class  f
7 vg . . . . . . 7  setvar  g
87cv 1361 . . . . . 6  class  g
92cv 1361 . . . . . . 7  class  c
103cv 1361 . . . . . . 7  class  d
11 cfunc 14746 . . . . . . 7  class  Func
129, 10, 11co 6080 . . . . . 6  class  ( c 
Func  d )
136, 8, 12wbr 4280 . . . . 5  wff  f ( c  Func  d )
g
14 vx . . . . . . . . . . 11  setvar  x
1514cv 1361 . . . . . . . . . 10  class  x
16 vy . . . . . . . . . . 11  setvar  y
1716cv 1361 . . . . . . . . . 10  class  y
1815, 17, 8co 6080 . . . . . . . . 9  class  ( x g y )
1918crn 4828 . . . . . . . 8  class  ran  (
x g y )
2015, 6cfv 5406 . . . . . . . . 9  class  ( f `
 x )
2117, 6cfv 5406 . . . . . . . . 9  class  ( f `
 y )
22 chom 14231 . . . . . . . . . 10  class  Hom
2310, 22cfv 5406 . . . . . . . . 9  class  ( Hom  `  d )
2420, 21, 23co 6080 . . . . . . . 8  class  ( ( f `  x ) ( Hom  `  d
) ( f `  y ) )
2519, 24wceq 1362 . . . . . . 7  wff  ran  (
x g y )  =  ( ( f `
 x ) ( Hom  `  d )
( f `  y
) )
26 cbs 14156 . . . . . . . 8  class  Base
279, 26cfv 5406 . . . . . . 7  class  ( Base `  c )
2825, 16, 27wral 2705 . . . . . 6  wff  A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) ( Hom  `  d ) ( f `
 y ) )
2928, 14, 27wral 2705 . . . . 5  wff  A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) ( Hom  `  d ) ( f `
 y ) )
3013, 29wa 369 . . . 4  wff  ( f ( c  Func  d
) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) ( Hom  `  d ) ( f `
 y ) ) )
3130, 5, 7copab 4337 . . 3  class  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) ( Hom  `  d ) ( f `
 y ) ) ) }
322, 3, 4, 4, 31cmpt2 6082 . 2  class  ( c  e.  Cat ,  d  e.  Cat  |->  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) ( Hom  `  d ) ( f `
 y ) ) ) } )
331, 32wceq 1362 1  wff Full  =  ( c  e.  Cat , 
d  e.  Cat  |->  {
<. f ,  g >.  |  ( f ( c  Func  d )
g  /\  A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) ( Hom  `  d ) ( f `
 y ) ) ) } )
Colors of variables: wff setvar class
This definition is referenced by:  fullfunc  14798  isfull  14802
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