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Definition df-full 13794
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 13792 . 2  class Full
2 vc . . 3  set  c
3 vd . . 3  set  d
4 ccat 13582 . . 3  class  Cat
5 vf . . . . . . 7  set  f
65cv 1631 . . . . . 6  class  f
7 vg . . . . . . 7  set  g
87cv 1631 . . . . . 6  class  g
92cv 1631 . . . . . . 7  class  c
103cv 1631 . . . . . . 7  class  d
11 cfunc 13744 . . . . . . 7  class  Func
129, 10, 11co 5874 . . . . . 6  class  ( c 
Func  d )
136, 8, 12wbr 4039 . . . . 5  wff  f ( c  Func  d )
g
14 vx . . . . . . . . . . 11  set  x
1514cv 1631 . . . . . . . . . 10  class  x
16 vy . . . . . . . . . . 11  set  y
1716cv 1631 . . . . . . . . . 10  class  y
1815, 17, 8co 5874 . . . . . . . . 9  class  ( x g y )
1918crn 4706 . . . . . . . 8  class  ran  (
x g y )
2015, 6cfv 5271 . . . . . . . . 9  class  ( f `
 x )
2117, 6cfv 5271 . . . . . . . . 9  class  ( f `
 y )
22 chom 13235 . . . . . . . . . 10  class  Hom
2310, 22cfv 5271 . . . . . . . . 9  class  (  Hom  `  d )
2420, 21, 23co 5874 . . . . . . . 8  class  ( ( f `  x ) (  Hom  `  d
) ( f `  y ) )
2519, 24wceq 1632 . . . . . . 7  wff  ran  (
x g y )  =  ( ( f `
 x ) (  Hom  `  d )
( f `  y
) )
26 cbs 13164 . . . . . . . 8  class  Base
279, 26cfv 5271 . . . . . . 7  class  ( Base `  c )
2825, 16, 27wral 2556 . . . . . 6  wff  A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  d ) ( f `
 y ) )
2928, 14, 27wral 2556 . . . . 5  wff  A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  d ) ( f `
 y ) )
3013, 29wa 358 . . . 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 4092 . . 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 5876 . 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 1632 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 set class
This definition is referenced by:  fullfunc  13796  isfull  13800
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