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Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrescbas 13701 Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   =>    |-  ( ph  ->  S  =  ( Base `  D ) )
 
Theoremreschom 13702 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   =>    |-  ( ph  ->  H  =  (  Hom  `  D ) )
 
Theoremreschomf 13703 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   =>    |-  ( ph  ->  H  =  (  Homf  `  D ) )
 
Theoremrescco 13704 Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   &    |-  .x.  =  (comp `  C )   =>    |-  ( ph  ->  .x. 
 =  (comp `  D ) )
 
Theoremrescabs 13705 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  J  Fn  ( T  X.  T ) )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  T 
 C_  S )   =>    |-  ( ph  ->  ( ( C  |`cat  H )  |`cat  J )  =  ( C  |`cat  J ) )
 
Theoremrescabs2 13706 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  J  Fn  ( T  X.  T ) )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  T  C_  S )   =>    |-  ( ph  ->  (
 ( Cs  S )  |`cat  J )  =  ( C  |`cat  J )
 )
 
Theoremissubc 13707* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  H  =  (  Homf  `  C )   &    |- 
 .1.  =  ( Id `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  S  =  dom  dom 
 J )   =>    |-  ( ph  ->  ( J  e.  (Subcat `  C ) 
 <->  ( J  C_cat  H  /\  A. x  e.  S  ( (  .1.  `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
 <. x ,  y >.  .x.  z ) f )  e.  ( x J z ) ) ) ) )
 
Theoremissubc2 13708* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  H  =  (  Homf  `  C )   &    |- 
 .1.  =  ( Id `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  H  /\  A. x  e.  S  ( (  .1.  `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
 <. x ,  y >.  .x.  z ) f )  e.  ( x J z ) ) ) ) )
 
Theoremsubcssc 13709 An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  H  =  (  Homf  `  C )   =>    |-  ( ph  ->  J  C_cat  H )
 
Theoremsubcfn 13710 An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  S  =  dom  dom  J )   =>    |-  ( ph  ->  J  Fn  ( S  X.  S ) )
 
Theoremsubcss1 13711 The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  B  =  (
 Base `  C )   =>    |-  ( ph  ->  S 
 C_  B )
 
Theoremsubcss2 13712 The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  ( X J Y )  C_  ( X H Y ) )
 
Theoremsubcidcl 13713 The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  .1.  =  ( Id `  C )   =>    |-  ( ph  ->  (  .1.  `  X )  e.  ( X J X ) )
 
Theoremsubccocl 13714 A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  Y  e.  S )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ph  ->  F  e.  ( X J Y ) )   &    |-  ( ph  ->  G  e.  ( Y J Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) )
 
Theoremsubccatid 13715* A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  .1.  =  ( Id `  C )   =>    |-  ( ph  ->  ( D  e.  Cat  /\  ( Id `  D )  =  ( x  e.  S  |->  (  .1.  `  x )
 ) ) )
 
Theoremsubcid 13716 The identity in a subcategory is the same as the original category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  X  e.  S )   =>    |-  ( ph  ->  (  .1.  `  X )  =  ( ( Id `  D ) `  X ) )
 
Theoremsubccat 13717 A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   =>    |-  ( ph  ->  D  e.  Cat )
 
Theoremissubc3 13718* Alternate definition of a subcategory, as a subset of the category which is itself a category. The assumption that the identity be closed is necessary just as in the case of a monoid, issubm2 14421, for the same reasons, since categories are a generalization of monoids. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  H  =  (  Homf  `  C )   &    |- 
 .1.  =  ( Id `  C )   &    |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  H  /\  A. x  e.  S  (  .1.  `  x )  e.  ( x J x )  /\  D  e.  Cat ) ) )
 
Theoremfullsubc 13719 The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory). (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Homf  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  S  C_  B )   =>    |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  e.  (Subcat `  C )
 )
 
Theoremfullresc 13720 The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Homf  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  S  C_  B )   &    |-  D  =  ( Cs  S )   &    |-  E  =  ( C  |`cat  ( H  |`  ( S  X.  S ) ) )   =>    |-  ( ph  ->  (
 (  Homf  `  D )  =  ( 
 Homf  `  E )  /\  (compf `  D )  =  (compf `  E ) ) )
 
Theoremresscat 13721 A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ( C  e.  Cat  /\  S  e.  V ) 
 ->  ( Cs  S )  e.  Cat )
 
Theoremsubsubc 13722 A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   =>    |-  ( H  e.  (Subcat `  C )  ->  ( J  e.  (Subcat `  D ) 
 <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H ) ) )
 
8.1.6  Functors
 
Syntaxcfunc 13723 Extend class notation with the class of all functors.
 class  Func
 
Syntaxcidfu 13724 Extend class notation with identity functor.
 class idfunc
 
Syntaxccofu 13725 Extend class notation with functor composition.
 class  o.func
 
Syntaxcresf 13726 Extend class notation to include restriction of a functor to a subcategory.
 class  |`f
 
Definitiondf-func 13727* Function returning all the functors from a category  t to a category  u. Intuitively a functor associates any morphism of  t to a morphism of  u, any object of  t to an object of  u, and respects the identity, the composition, the domain and the codomain. Here to capture the idea that a functor associates any object of  t to an object of  u we write it associates any identity of  t to an identity of  u which simplifies the definition. (Contributed by FL, 10-Feb-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
 |- 
 Func  =  ( t  e.  Cat ,  u  e. 
 Cat  |->  { <. f ,  g >.  |  [. ( Base `  t )  /  b ]. ( f : b --> ( Base `  u )  /\  g  e.  X_ z  e.  ( b  X.  b
 ) ( ( ( f `  ( 1st `  z ) ) ( 
 Hom  `  u )
 ( f `  ( 2nd `  z ) ) )  ^m  ( ( 
 Hom  `  t ) `  z ) )  /\  A. x  e.  b  ( ( ( x g x ) `  (
 ( Id `  t
 ) `  x )
 )  =  ( ( Id `  u ) `
  ( f `  x ) )  /\  A. y  e.  b  A. z  e.  b  A. m  e.  ( x (  Hom  `  t )
 y ) A. n  e.  ( y (  Hom  `  t ) z ) ( ( x g z ) `  ( n ( <. x ,  y >. (comp `  t
 ) z ) m ) )  =  ( ( ( y g z ) `  n ) ( <. ( f `
  x ) ,  ( f `  y
 ) >. (comp `  u ) ( f `  z ) ) ( ( x g y ) `  m ) ) ) ) }
 )
 
Definitiondf-idfu 13728* Define the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- idfunc  =  ( t  e.  Cat  |->  [_ ( Base `  t )  /  b ]_ <. (  _I  |`  b ) ,  (
 z  e.  ( b  X.  b )  |->  (  _I  |`  ( (  Hom  `  t ) `  z ) ) )
 >. )
 
Definitiondf-cofu 13729* Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- 
 o.func  =  ( g  e.  _V ,  f  e.  _V  |->  <.
 ( ( 1st `  g
 )  o.  ( 1st `  f ) ) ,  ( x  e.  dom  dom  ( 2nd `  f
 ) ,  y  e. 
 dom  dom  ( 2nd `  f
 )  |->  ( ( ( ( 1st `  f
 ) `  x )
 ( 2nd `  g )
 ( ( 1st `  f
 ) `  y )
 )  o.  ( x ( 2nd `  f
 ) y ) ) ) >. )
 
Definitiondf-resf 13730* Define the restriction of a functor to a subcategory (analogue of df-res 4699). (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  |`f  =  ( f  e.  _V ,  h  e.  _V  |->  <.
 ( ( 1st `  f
 )  |`  dom  dom  h ) ,  ( x  e. 
 dom  h  |->  ( ( ( 2nd `  f
 ) `  x )  |`  ( h `  x ) ) ) >. )
 
Theoremrelfunc 13731 The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 Rel  ( D  Func  E )
 
Theoremfuncrcl 13732 Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( F  e.  ( D  Func  E )  ->  ( D  e.  Cat  /\  E  e.  Cat )
 )
 
Theoremisfunc 13733* Value of the set of functors between two categories. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  C  =  (
 Base `  E )   &    |-  H  =  (  Hom  `  D )   &    |-  J  =  (  Hom  `  E )   &    |-  .1.  =  ( Id `  D )   &    |-  I  =  ( Id `  E )   &    |-  .x.  =  (comp `  D )   &    |-  O  =  (comp `  E )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   =>    |-  ( ph  ->  ( F ( D  Func  E ) G  <->  ( F : B
 --> C  /\  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `  ( 1st `  z )
 ) J ( F `
  ( 2nd `  z
 ) ) )  ^m  ( H `  z ) )  /\  A. x  e.  B  ( ( ( x G x ) `
  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x H y ) A. n  e.  ( y H z ) ( ( x G z ) `  ( n ( <. x ,  y >.  .x.  z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) ) ) ) ) )
 
Theoremisfuncd 13734* Deduce that an operation is a functor of categories. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  C  =  (
 Base `  E )   &    |-  H  =  (  Hom  `  D )   &    |-  J  =  (  Hom  `  E )   &    |-  .1.  =  ( Id `  D )   &    |-  I  =  ( Id `  E )   &    |-  .x.  =  (comp `  D )   &    |-  O  =  (comp `  E )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   &    |-  ( ph  ->  F : B --> C )   &    |-  ( ph  ->  G  Fn  ( B  X.  B ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x G y ) : ( x H y ) --> ( ( F `
  x ) J ( F `  y
 ) ) )   &    |-  (
 ( ph  /\  x  e.  B )  ->  (
 ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  /\  ( m  e.  ( x H y )  /\  n  e.  ( y H z ) ) )  ->  ( ( x G z ) `  ( n ( <. x ,  y >.  .x.  z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) ) )   =>    |-  ( ph  ->  F ( D  Func  E ) G )
 
Theoremfuncf1 13735 The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  C  =  (
 Base `  E )   &    |-  ( ph  ->  F ( D 
 Func  E ) G )   =>    |-  ( ph  ->  F : B
 --> C )
 
Theoremfuncixp 13736* The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  H  =  ( 
 Hom  `  D )   &    |-  J  =  (  Hom  `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   =>    |-  ( ph  ->  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `  ( 1st `  z )
 ) J ( F `
  ( 2nd `  z
 ) ) )  ^m  ( H `  z ) ) )
 
Theoremfuncf2 13737 The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  H  =  ( 
 Hom  `  D )   &    |-  J  =  (  Hom  `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X G Y ) : ( X H Y )
 --> ( ( F `  X ) J ( F `  Y ) ) )
 
Theoremfuncfn2 13738 The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  ( ph  ->  F ( D  Func  E ) G )   =>    |-  ( ph  ->  G  Fn  ( B  X.  B ) )
 
Theoremfuncid 13739 A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  .1.  =  ( Id `  D )   &    |-  I  =  ( Id `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( ( X G X ) `  (  .1.  `  X ) )  =  ( I `  ( F `  X ) ) )
 
Theoremfuncco 13740 A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  H  =  ( 
 Hom  `  D )   &    |-  .x.  =  (comp `  D )   &    |-  O  =  (comp `  E )   &    |-  ( ph  ->  F ( D 
 Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  M  e.  ( X H Y ) )   &    |-  ( ph  ->  N  e.  ( Y H Z ) )   =>    |-  ( ph  ->  ( ( X G Z ) `  ( N (
 <. X ,  Y >.  .x. 
 Z ) M ) )  =  ( ( ( Y G Z ) `  N ) (
 <. ( F `  X ) ,  ( F `  Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) )
 
Theoremfuncsect 13741 The image of a section under a functor is a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  S  =  (Sect `  D )   &    |-  T  =  (Sect `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M ( X S Y ) N )   =>    |-  ( ph  ->  (
 ( X G Y ) `  M ) ( ( F `  X ) T ( F `  Y ) ) ( ( Y G X ) `  N ) )
 
Theoremfuncinv 13742 The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  I  =  (Inv `  D )   &    |-  J  =  (Inv `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M ( X I Y ) N )   =>    |-  ( ph  ->  (
 ( X G Y ) `  M ) ( ( F `  X ) J ( F `  Y ) ) ( ( Y G X ) `  N ) )
 
Theoremfunciso 13743 The image of an isomorphism under a functor is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  I  =  ( 
 Iso  `  D )   &    |-  J  =  (  Iso  `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M  e.  ( X I Y ) )   =>    |-  ( ph  ->  (
 ( X G Y ) `  M )  e.  ( ( F `  X ) J ( F `  Y ) ) )
 
Theoremfuncoppc 13744 A functor on categories yields a functor on the opposite categories (in the same direction). (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  P  =  (oppCat `  D )   &    |-  ( ph  ->  F ( C  Func  D ) G )   =>    |-  ( ph  ->  F ( O  Func  P )tpos 
 G )
 
Theoremidfuval 13745* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  H  =  (  Hom  `  C )   =>    |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `
  z ) ) ) >. )
 
Theoremidfu2nd 13746 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X ( 2nd `  I
 ) Y )  =  (  _I  |`  ( X H Y ) ) )
 
Theoremidfu2 13747 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( ( X ( 2nd `  I
 ) Y ) `  F )  =  F )
 
Theoremidfu1st 13748 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   =>    |-  ( ph  ->  ( 1st `  I )  =  (  _I  |`  B ) )
 
Theoremidfu1 13749 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (
 ( 1st `  I ) `  X )  =  X )
 
Theoremidfucl 13750 The identity functor is a functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   =>    |-  ( C  e.  Cat  ->  I  e.  ( C  Func  C ) )
 
Theoremcofuval 13751* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   =>    |-  ( ph  ->  ( G  o.func 
 F )  =  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `  x ) ( 2nd `  G )
 ( ( 1st `  F ) `  y ) )  o.  ( x ( 2nd `  F )
 y ) ) )
 >. )
 
Theoremcofu1st 13752 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   =>    |-  ( ph  ->  ( 1st `  ( G  o.func  F ) )  =  ( ( 1st `  G )  o.  ( 1st `  F ) ) )
 
Theoremcofu1 13753 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( ( 1st `  ( G  o.func 
 F ) ) `  X )  =  (
 ( 1st `  G ) `  ( ( 1st `  F ) `  X ) ) )
 
Theoremcofu2nd 13754 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X ( 2nd `  ( G  o.func 
 F ) ) Y )  =  ( ( ( ( 1st `  F ) `  X ) ( 2nd `  G )
 ( ( 1st `  F ) `  Y ) )  o.  ( X ( 2nd `  F ) Y ) ) )
 
Theoremcofu2 13755 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( ( X ( 2nd `  ( G  o.func  F )
 ) Y ) `  R )  =  (
 ( ( ( 1st `  F ) `  X ) ( 2nd `  G ) ( ( 1st `  F ) `  Y ) ) `  (
 ( X ( 2nd `  F ) Y ) `
  R ) ) )
 
Theoremcofuval2 13756* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F ( C  Func  D ) G )   &    |-  ( ph  ->  H ( D  Func  E ) K )   =>    |-  ( ph  ->  ( <. H ,  K >.  o.func  <. F ,  G >. )  = 
 <. ( H  o.  F ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( F `  x ) K ( F `  y ) )  o.  ( x G y ) ) ) >. )
 
Theoremcofucl 13757 The composition of two functors is a functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   =>    |-  ( ph  ->  ( G  o.func 
 F )  e.  ( C  Func  E ) )
 
Theoremcofuass 13758 Functor composition is associative. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  ( ph  ->  H  e.  ( D  Func  E ) )   &    |-  ( ph  ->  K  e.  ( E  Func  F ) )   =>    |-  ( ph  ->  (
 ( K  o.func  H )  o.func  G )  =  ( K  o.func  ( H  o.func  G )
 ) )
 
Theoremcofulid 13759 The identity functor is a left identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  I  =  (idfunc `  D )   =>    |-  ( ph  ->  ( I  o.func 
 F )  =  F )
 
Theoremcofurid 13760 The identity functor is a right identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  I  =  (idfunc `  C )   =>    |-  ( ph  ->  ( F  o.func 
 I )  =  F )
 
Theoremresfval 13761* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  H  e.  W )   =>    |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F )  |`  dom  dom  H ) ,  ( x  e. 
 dom  H  |->  ( ( ( 2nd `  F ) `  x )  |`  ( H `  x ) ) ) >. )
 
Theoremresfval2 13762* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  H  e.  W )   &    |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( <. F ,  G >.  |`f  H )  =  <. ( F  |`  S ) ,  ( x  e.  S ,  y  e.  S  |->  ( ( x G y )  |`  ( x H y ) ) ) >. )
 
Theoremresf1st 13763 Value of the functor restriction operator on objects. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  H  e.  W )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( 1st `  ( F  |`f  H ) )  =  ( ( 1st `  F )  |`  S ) )
 
Theoremresf2nd 13764 Value of the functor restriction operator on morphisms. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  H  e.  W )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  ( X ( 2nd `  ( F  |`f  H ) ) Y )  =  ( ( X ( 2nd `  F ) Y )  |`  ( X H Y ) ) )
 
Theoremfuncres 13765 A functor restricted to a subcategory is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  H  e.  (Subcat `  C ) )   =>    |-  ( ph  ->  ( F  |`f  H )  e.  (
 ( C  |`cat  H )  Func  D ) )
 
Theoremfuncres2b 13766* Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  A  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  R  e.  (Subcat `  D ) )   &    |-  ( ph  ->  R  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  F : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A ) )  ->  ( x G y ) : Y --> ( ( F `
  x ) R ( F `  y
 ) ) )   =>    |-  ( ph  ->  ( F ( C  Func  D ) G  <->  F ( C  Func  ( D  |`cat  R ) ) G ) )
 
Theoremfuncres2 13767 A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( R  e.  (Subcat `  D )  ->  ( C  Func  ( D  |`cat  R )
 )  C_  ( C  Func  D ) )
 
Theoremwunfunc 13768 A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  C  e.  U )   &    |-  ( ph  ->  D  e.  U )   =>    |-  ( ph  ->  ( C  Func  D )  e.  U )
 
Theoremfuncpropd 13769 If two categories have the same set of objects, morphisms, and compositions, then they have the same functors. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  A )  =  ( 
 Homf  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  (  Homf  `  C )  =  (  Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  ( A  Func  C )  =  ( B  Func  D ) )
 
Theoremfuncres2c 13770 Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
 |-  A  =  ( Base `  C )   &    |-  E  =  ( Ds  S )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  F : A --> S )   =>    |-  ( ph  ->  ( F ( C  Func  D ) G  <->  F ( C  Func  E ) G ) )
 
8.1.7  Full & faithful functors
 
Syntaxcful 13771 Extend class notation with the class of all full functors.
 class Full
 
Syntaxcfth 13772 Extend class notation with the class of all faithful functors.
 class Faith
 
Definitiondf-full 13773* 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.)
 |- 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 ) ) ) } )
 
Definitiondf-fth 13774* 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.)
 |- 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 ) ) } )
 
Theoremfullfunc 13775 A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( C Full  D ) 
 C_  ( C  Func  D )
 
Theoremfthfunc 13776 A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( C Faith  D ) 
 C_  ( C  Func  D )
 
Theoremrelfull 13777 The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |- 
 Rel  ( C Full  D )
 
Theoremrelfth 13778 The set of faithful functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |- 
 Rel  ( C Faith  D )
 
Theoremisfull 13779* Value of the set of full functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  J  =  ( 
 Hom  `  D )   =>    |-  ( F ( C Full  D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  ran  ( x G y )  =  ( ( F `  x ) J ( F `  y ) ) ) )
 
Theoremisfull2 13780* Equivalent condition for a full functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  J  =  ( 
 Hom  `  D )   &    |-  H  =  (  Hom  `  C )   =>    |-  ( F ( C Full 
 D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y )
 -onto-> ( ( F `  x ) J ( F `  y ) ) ) )
 
Theoremfullfo 13781 The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  J  =  ( 
 Hom  `  D )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  F ( C Full  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X G Y ) : ( X H Y ) -onto-> ( ( F `
  X ) J ( F `  Y ) ) )
 
Theoremfulli 13782* The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  J  =  ( 
 Hom  `  D )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  F ( C Full  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( ( F `  X ) J ( F `  Y ) ) )   =>    |-  ( ph  ->  E. f  e.  ( X H Y ) R  =  (
 ( X G Y ) `  f ) )
 
Theoremisfth 13783* Value of the set of faithful functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   =>    |-  ( F ( C Faith  D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  Fun  `' ( x G y ) ) )
 
Theoremisfth2 13784* Equivalent condition for a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  J  =  (  Hom  `  D )   =>    |-  ( F ( C Faith  D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y )
 -1-1-> ( ( F `  x ) J ( F `  y ) ) ) )
 
Theoremisffth2 13785* A fully faithful functor is a functor which is bijective on hom-sets. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  J  =  (  Hom  `  D )   =>    |-  ( F ( ( C Full  D )  i^i  ( C Faith  D ) ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y ) -1-1-onto-> ( ( F `  x ) J ( F `  y ) ) ) )
 
Theoremfthf1 13786 The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-> ( ( F `
  X ) J ( F `  Y ) ) )
 
Theoremfthi 13787 The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   &    |-  ( ph  ->  S  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( ( ( X G Y ) `  R )  =  ( ( X G Y ) `  S )  <->  R  =  S ) )
 
Theoremffthf1o 13788 The morphism map of a fully faithful functor is a bijection. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  ( ph  ->  F ( ( C Full  D )  i^i  ( C Faith  D ) ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-onto-> ( ( F `  X ) J ( F `  Y ) ) )
 
Theoremfullpropd 13789 If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  A )  =  ( 
 Homf  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  (  Homf  `  C )  =  (  Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  ( A Full  C )  =  ( B Full  D ) )
 
Theoremfthpropd 13790 If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  A )  =  ( 
 Homf  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  (  Homf  `  C )  =  (  Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  ( A Faith  C )  =  ( B Faith  D ) )
 
Theoremfulloppc 13791 The opposite functor of a full functor is also full. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  P  =  (oppCat `  D )   &    |-  ( ph  ->  F ( C Full  D ) G )   =>    |-  ( ph  ->  F ( O Full  P )tpos  G )
 
Theoremfthoppc 13792 The opposite functor of a faithful functor is also faithful. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  P  =  (oppCat `  D )   &    |-  ( ph  ->  F ( C Faith  D ) G )   =>    |-  ( ph  ->  F ( O Faith  P )tpos  G )
 
Theoremffthoppc 13793 The opposite functor of a fully faithful functor is also full and faithful. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  P  =  (oppCat `  D )   &    |-  ( ph  ->  F ( ( C Full  D )  i^i  ( C Faith  D ) ) G )   =>    |-  ( ph  ->  F (
 ( O Full  P )  i^i  ( O Faith  P ) )tpos  G )
 
Theoremfthsect 13794 A faithful functor reflects sections. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M  e.  ( X H Y ) )   &    |-  ( ph  ->  N  e.  ( Y H X ) )   &    |-  S  =  (Sect `  C )   &    |-  T  =  (Sect `  D )   =>    |-  ( ph  ->  ( M ( X S Y ) N  <->  ( ( X G Y ) `  M ) ( ( F `  X ) T ( F `  Y ) ) ( ( Y G X ) `  N ) ) )
 
Theoremfthinv 13795 A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M  e.  ( X H Y ) )   &    |-  ( ph  ->  N  e.  ( Y H X ) )   &    |-  I  =  (Inv `  C )   &    |-  J  =  (Inv `  D )   =>    |-  ( ph  ->  ( M ( X I Y ) N  <->  ( ( X G Y ) `  M ) ( ( F `  X ) J ( F `  Y ) ) ( ( Y G X ) `  N ) ) )
 
Theoremfthmon 13796 A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   &    |-  M  =  (Mono `  C )   &    |-  N  =  (Mono `  D )   &    |-  ( ph  ->  ( ( X G Y ) `  R )  e.  ( ( F `  X ) N ( F `  Y ) ) )   =>    |-  ( ph  ->  R  e.  ( X M Y ) )
 
Theoremfthepi 13797 A faithful functor reflects epimorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   &    |-  E  =  (Epi `  C )   &    |-  P  =  (Epi `  D )   &    |-  ( ph  ->  ( ( X G Y ) `  R )  e.  ( ( F `  X ) P ( F `  Y ) ) )   =>    |-  ( ph  ->  R  e.  ( X E Y ) )
 
Theoremffthiso 13798 A fully faithful functor reflects isomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   &    |-  ( ph  ->  F ( C Full  D ) G )   &    |-  I  =  (  Iso  `  C )   &    |-  J  =  ( 
 Iso  `  D )   =>    |-  ( ph  ->  ( R  e.  ( X I Y )  <-> 
 ( ( X G Y ) `  R )  e.  ( ( F `  X ) J ( F `  Y ) ) ) )
 
Theoremfthres2b 13799* Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  A  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  R  e.  (Subcat `  D ) )   &    |-  ( ph  ->  R  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  F : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A ) )  ->  ( x G y ) : Y --> ( ( F `
  x ) R ( F `  y
 ) ) )   =>    |-  ( ph  ->  ( F ( C Faith  D ) G  <->  F ( C Faith  ( D  |`cat  R ) ) G ) )
 
Theoremfthres2c 13800 Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
 |-  A  =  ( Base `  C )   &    |-  E  =  ( Ds  S )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  F : A --> S )   =>    |-  ( ph  ->  ( F ( C Faith  D ) G  <->  F ( C Faith  E ) G ) )
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