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Definition df-fth 13779
Description: Function returning all the faithful 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 injections. (Contributed by Mario Carneiro, 26-Jan-2017.)
Assertion
Ref Expression
df-fth  |- Faith  =  ( c  e.  Cat , 
d  e.  Cat  |->  {
<. f ,  g >.  |  ( f ( c  Func  d )
g  /\  A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) ) } )
Distinct variable group:    c, d, f, g, x, y

Detailed syntax breakdown of Definition df-fth
StepHypRef Expression
1 cfth 13777 . 2  class Faith
2 vc . . 3  set  c
3 vd . . 3  set  d
4 ccat 13566 . . 3  class  Cat
5 vf . . . . . . 7  set  f
65cv 1622 . . . . . 6  class  f
7 vg . . . . . . 7  set  g
87cv 1622 . . . . . 6  class  g
92cv 1622 . . . . . . 7  class  c
103cv 1622 . . . . . . 7  class  d
11 cfunc 13728 . . . . . . 7  class  Func
129, 10, 11co 5858 . . . . . 6  class  ( c 
Func  d )
136, 8, 12wbr 4023 . . . . 5  wff  f ( c  Func  d )
g
14 vx . . . . . . . . . . 11  set  x
1514cv 1622 . . . . . . . . . 10  class  x
16 vy . . . . . . . . . . 11  set  y
1716cv 1622 . . . . . . . . . 10  class  y
1815, 17, 8co 5858 . . . . . . . . 9  class  ( x g y )
1918ccnv 4688 . . . . . . . 8  class  `' ( x g y )
2019wfun 5249 . . . . . . 7  wff  Fun  `' ( x g y )
21 cbs 13148 . . . . . . . 8  class  Base
229, 21cfv 5255 . . . . . . 7  class  ( Base `  c )
2320, 16, 22wral 2543 . . . . . 6  wff  A. y  e.  ( Base `  c
) Fun  `' (
x g y )
2423, 14, 22wral 2543 . . . . 5  wff  A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) Fun  `' (
x g y )
2513, 24wa 358 . . . 4  wff  ( f ( c  Func  d
) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) )
2625, 5, 7copab 4076 . . 3  class  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) ) }
272, 3, 4, 4, 26cmpt2 5860 . 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
) Fun  `' (
x g y ) ) } )
281, 27wceq 1623 1  wff Faith  =  ( c  e.  Cat , 
d  e.  Cat  |->  {
<. f ,  g >.  |  ( f ( c  Func  d )
g  /\  A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) ) } )
Colors of variables: wff set class
This definition is referenced by:  fthfunc  13781  isfth  13788
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