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Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhomgrf 25201 Homset of a composite. JFM CAT1 th. 51 (Contributed by FL, 10-Mar-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( A  e.  O  /\  B  e.  O  /\  C  e.  O ) )  ->  ( ( F  e.  ( H ` 
 <. A ,  B >. ) 
 /\  G  e.  ( H `  <. B ,  C >. ) )  ->  ( G R F )  e.  ( H `  <. A ,  C >. ) ) )
 
18.12.50  Monomorphisms, Epimorphisms, Isomorphisms
 
SyntaxcepiOLD 25202 Extend class notation with the class of all epimorphisms.
 class Epic
 
SyntaxcmonOLD 25203 Extend class notation with the class of all monomorphisms.
 class MonoOLD
 
SyntaxcisoOLD 25204 Extend class notation with the class of all isomorphisms.
 class  Iso OLD
 
Definitiondf-monOLD 25205* Function returning the monomorphisms of the category  x. JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.)
 |- MonoOLD  =  ( x  e.  Cat OLD  |->  { f  e.  dom  ( dom_ `  x )  |  A. g  e. 
 dom  ( dom_ `  x ) A. h  e.  dom  ( dom_ `  x )
 ( ( ( (
 dom_ `  x ) `  g )  =  (
 ( dom_ `  x ) `  h )  /\  (
 ( cod_ `  x ) `  g )  =  ( ( dom_ `  x ) `  f )  /\  (
 ( cod_ `  x ) `  h )  =  ( ( dom_ `  x ) `  f ) )  ->  ( ( f ( o_ `  x ) g )  =  ( f ( o_ `  x ) h ) 
 ->  g  =  h ) ) } )
 
Definitiondf-epiOLD 25206* Function returning the epimorphisms of the category  x. JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.)
 |- Epic  =  ( x  e.  Cat OLD  |->  { f  e.  dom  ( dom_ `  x )  | 
 A. g  e.  dom  ( dom_ `  x ) A. h  e.  dom  ( dom_ `  x )
 ( ( ( (
 cod_ `  x ) `  g )  =  (
 ( cod_ `  x ) `  h )  /\  (
 ( dom_ `  x ) `  g )  =  ( ( cod_ `  x ) `  f )  /\  (
 ( dom_ `  x ) `  h )  =  ( ( cod_ `  x ) `  f ) )  ->  ( ( g ( o_ `  x ) f )  =  ( h ( o_ `  x ) f ) 
 ->  g  =  h ) ) } )
 
Definitiondf-isoc 25207* Function returning the isomorphisms of the category  x. The Joy of Cats p. 28. (Contributed by FL, 9-Jun-2014.)
 |-  Iso OLD 
 =  ( x  e. 
 Cat OLD  |->  { f  e.  dom  ( dom_ `  x )  |  E. g  e.  dom  ( dom_ `  x )
 ( ( ( dom_ `  x ) `  g
 )  =  ( (
 cod_ `  x ) `  f )  /\  ( (
 cod_ `  x ) `  g )  =  (
 ( dom_ `  x ) `  f )  /\  (
 ( f ( o_
 `  x ) g )  =  ( ( id_ `  x ) `  ( ( dom_ `  x ) `  g ) ) 
 /\  ( g ( o_ `  x ) f )  =  ( ( id_ `  x ) `  ( ( dom_ `  x ) `  f
 ) ) ) ) } )
 
Theoremismona 25208* Monomorphisms of a category. (Contributed by FL, 5-Dec-2007.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  ( MonoOLD  `  T )  =  { f  e.  M  |  A. g  e.  M  A. h  e.  M  ( ( ( D `  g )  =  ( D `  h )  /\  ( C `  g )  =  ( D `  f )  /\  ( C `
  h )  =  ( D `  f
 ) )  ->  (
 ( f R g )  =  ( f R h )  ->  g  =  h )
 ) } )
 
Theoremismonb 25209* The predicate "is a monomorphism". (Contributed by FL, 2-Jan-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  ( F  e.  ( MonoOLD  `  T ) 
 <->  ( F  e.  M  /\  A. g  e.  M  A. h  e.  M  ( ( ( D `  g )  =  ( D `  h )  /\  ( C `  g )  =  ( D `  F )  /\  ( C `
  h )  =  ( D `  F ) )  ->  ( ( F R g )  =  ( F R h )  ->  g  =  h ) ) ) ) )
 
Theoremismonb1 25210* The predicate "is a monomorphism". (Contributed by FL, 2-Jan-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  F  e.  M )  ->  ( F  e.  ( MonoOLD  `  T )  <->  A. g  e.  M  A. h  e.  M  ( ( ( D `  g )  =  ( D `  h )  /\  ( C `  g )  =  ( D `  F )  /\  ( C `
  h )  =  ( D `  F ) )  ->  ( ( F R g )  =  ( F R h )  ->  g  =  h ) ) ) )
 
Theoremismonb2 25211 A monomorphism is a left-cancelable morphism. (Contributed by FL, 2-Jan-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( F  e.  M  /\  G  e.  M  /\  J  e.  M ) )  ->  ( F  e.  ( MonoOLD  `  T )  ->  ( ( ( D `
  G )  =  ( D `  J )  /\  ( C `  G )  =  ( D `  F )  /\  ( C `  J )  =  ( D `  F ) )  ->  ( ( F R G )  =  ( F R J )  ->  G  =  J )
 ) ) )
 
Theoremimonclem 25212* Lemma for ismonc 25213. (Contributed by FL, 1-Jan-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( B  e.  O  /\  C  e.  O )  /\  F  e.  ( H `  <. B ,  C >. ) )  ->  ( A. a  e.  O  A. g  e.  ( H `
  <. a ,  B >. ) A. h  e.  ( H `  <. a ,  B >. ) ( ( F R g )  =  ( F R h )  ->  g  =  h )  ->  ( F  e.  dom  ( dom_ `  T )  /\  A. g  e.  dom  ( dom_ `  T ) A. h  e.  dom  ( dom_ `  T ) ( ( ( ( dom_ `  T ) `  g )  =  ( ( dom_ `  T ) `  h )  /\  (
 ( cod_ `  T ) `  g )  =  ( ( dom_ `  T ) `  F )  /\  (
 ( cod_ `  T ) `  h )  =  ( ( dom_ `  T ) `  F ) )  ->  ( ( F R g )  =  ( F R h )  ->  g  =  h )
 ) ) ) )
 
Theoremismonc 25213* The predicate "is a monomorphism" when the morphism belongs to a homset. (Contributed by FL, 2-Jan-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( B  e.  O  /\  C  e.  O )  /\  F  e.  ( H `  <. B ,  C >. ) )  ->  ( F  e.  ( MonoOLD  `  T )  <->  A. a  e.  O  A. g  e.  ( H `
  <. a ,  B >. ) A. h  e.  ( H `  <. a ,  B >. ) ( ( F R g )  =  ( F R h )  ->  g  =  h ) ) )
 
Theoremcmpmon 25214 The composite of two monomorphisms is a monomorphism. JFM CAT1 th. 61 (Contributed by FL, 10-Mar-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( ( A  e.  O  /\  B  e.  O  /\  C  e.  O )  /\  ( F  e.  ( H `  <. A ,  B >. ) 
 /\  G  e.  ( H `  <. B ,  C >. ) )  /\  ( F  e.  ( MonoOLD  `  T ) 
 /\  G  e.  ( MonoOLD  `  T ) ) ) ) 
 ->  ( G R F )  e.  ( MonoOLD  `  T ) )
 
Theoremicmpmon 25215 If  ( G R F ) is a monomorphism then  F is a monomorphism. JFM CAT1 th. 62 (Contributed by FL, 17-Mar-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( ( A  e.  O  /\  B  e.  O  /\  C  e.  O )  /\  ( F  e.  ( H `  <. A ,  B >. ) 
 /\  G  e.  ( H `  <. B ,  C >. ) )  /\  ( G R F )  e.  ( MonoOLD 
 `  T ) ) )  ->  F  e.  ( MonoOLD  `  T ) )
 
Theoremidmon 25216 If there exists  G such as  ( G R F )  =  ( J `  B ) then F is a monomorphism. JFM CAT1 th. 63. (Contributed by FL, 5-May-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  R  =  ( o_ `  T )   &    |-  J  =  ( id_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( A  e.  O  /\  B  e.  O )  /\  ( G  e.  ( H `  <. A ,  B >. )  /\  F  e.  ( H `  <. B ,  A >. ) ) ) 
 ->  ( ( G R F )  =  ( J `  B )  ->  F  e.  ( MonoOLD  `  T ) ) )
 
Theoremimmon 25217 A morphism identity is a monomorphism. JFM CAT1 th. 64. (Contributed by FL, 5-May-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  J  =  ( id_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O )  ->  ( J `  A )  e.  ( MonoOLD  `  T ) )
 
Theoremisepia 25218* Epimorphisms of a category  T. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  (Epic `  T )  =  { f  e.  M  |  A. g  e.  M  A. h  e.  M  ( ( ( C `  g )  =  ( C `  h )  /\  ( D `  g )  =  ( C `  f )  /\  ( D `
  h )  =  ( C `  f
 ) )  ->  (
 ( g R f )  =  ( h R f )  ->  g  =  h )
 ) } )
 
Theoremisepib 25219* The predicate "is an epimorphism". (Contributed by FL, 8-Aug-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  ( F  e.  (Epic `  T )  <->  ( F  e.  M  /\  A. g  e.  M  A. h  e.  M  ( ( ( C `  g )  =  ( C `  h )  /\  ( D `
  g )  =  ( C `  F )  /\  ( D `  h )  =  ( C `  F ) ) 
 ->  ( ( g R F )  =  ( h R F ) 
 ->  g  =  h ) ) ) ) )
 
Theoremisepib1 25220* The predicate "is an epimorphism". (Contributed by FL, 10-Aug-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  F  e.  M )  ->  ( F  e.  (Epic `  T )  <->  A. g  e.  M  A. h  e.  M  ( ( ( C `  g )  =  ( C `  h )  /\  ( D `  g )  =  ( C `  F )  /\  ( D `
  h )  =  ( C `  F ) )  ->  ( ( g R F )  =  ( h R F )  ->  g  =  h ) ) ) )
 
Theoremisepib2 25221 An epimorphism is a right-cancelable morphism. (Contributed by FL, 10-Aug-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( F  e.  M  /\  G  e.  M  /\  J  e.  M ) )  ->  ( F  e.  (Epic `  T )  ->  ( ( ( C `
  G )  =  ( C `  J )  /\  ( D `  G )  =  ( C `  F )  /\  ( D `  J )  =  ( C `  F ) )  ->  ( ( G R F )  =  ( J R F )  ->  G  =  J )
 ) ) )
 
Theoremiepiclem 25222* Lemma for isepic 25223. (Contributed by FL, 6-Oct-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( A  e.  O  /\  B  e.  O )  /\  F  e.  ( H `  <. A ,  B >. ) )  ->  ( A. c  e.  O  A. g  e.  ( H `
  <. B ,  c >. ) A. h  e.  ( H `  <. B ,  c >. ) ( ( g R F )  =  ( h R F )  ->  g  =  h )  ->  ( F  e.  dom  ( dom_ `  T )  /\  A. g  e.  dom  ( dom_ `  T ) A. h  e.  dom  ( dom_ `  T ) ( ( ( ( cod_ `  T ) `  g )  =  ( ( cod_ `  T ) `  h )  /\  (
 ( dom_ `  T ) `  g )  =  ( ( cod_ `  T ) `  F )  /\  (
 ( dom_ `  T ) `  h )  =  ( ( cod_ `  T ) `  F ) )  ->  ( ( g R F )  =  ( h R F ) 
 ->  g  =  h ) ) ) ) )
 
Theoremisepic 25223* The predicate "is an epimorphism" when the morphism belongs to a homset. (Contributed by FL, 27-Oct-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( A  e.  O  /\  B  e.  O )  /\  F  e.  ( H `  <. A ,  B >. ) )  ->  ( F  e.  (Epic `  T ) 
 <-> 
 A. c  e.  O  A. g  e.  ( H `
  <. B ,  c >. ) A. h  e.  ( H `  <. B ,  c >. ) ( ( g R F )  =  ( h R F )  ->  g  =  h ) ) )
 
Theoremisiso 25224* Isomorphisms of a category. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   &    |-  J  =  ( id_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  (  Iso OLD  `  T )  =  { f  e.  M  |  E. g  e.  M  ( ( D `
  g )  =  ( C `  f
 )  /\  ( C `  g )  =  ( D `  f ) 
 /\  ( ( f R g )  =  ( J `  ( D `  g ) ) 
 /\  ( g R f )  =  ( J `  ( D `
  f ) ) ) ) } )
 
Syntaxccinv 25225 Extend class notation to include a function that returns the inverses of a morphism.
 class  cinv OLD
 
Definitiondf-cinv 25226* Function ( indexed by the category 
x) returning the inverses of a morphism  f. (Contributed by FL, 22-Dec-2008.)
 |-  cinv OLD 
 =  ( x  e. 
 Cat OLD  |->  ( f  e. 
 dom  ( dom_ `  x )  |->  { g  e.  dom  ( dom_ `  x )  |  ( ( f ( o_ `  x ) g )  =  ( ( id_ `  x ) `  ( ( cod_ `  x ) `  f
 ) )  /\  (
 g ( o_ `  x ) f )  =  ( ( id_ `  x ) `  (
 ( dom_ `  x ) `  f ) ) ) } ) )
 
Theoremcinvlem1 25227* The set of the inverses of all the morphisms . (Contributed by FL, 22-Dec-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   &    |-  J  =  ( id_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  ( cinv OLD `  T )  =  ( f  e.  M  |->  { g  e.  M  |  ( ( f R g )  =  ( J `  ( C `
  f ) ) 
 /\  ( g R f )  =  ( J `  ( D `
  f ) ) ) } ) )
 
Theoremcinvlem2 25228* The set of the inverses of the morphism  F. (Contributed by FL, 22-Dec-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  T  e.  Cat OLD   =>    |-  ( F  e.  M  ->  ( ( cinv OLD `  T ) `  F )  =  {
 g  e.  M  |  ( ( F R g )  =  ( J `  ( C `  F ) )  /\  ( g R F )  =  ( J `  ( D `  F ) ) ) }
 )
 
Theoremcinvlem3 25229 The set of the inverses of the morphism  F. (Contributed by FL, 22-Dec-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  T  e.  Cat OLD   &    |-  F  e.  M   =>    |-  ( G  e.  (
 ( cinv OLD `  T ) `  F )  <->  ( G  e.  M  /\  ( F R G )  =  ( J `  ( C `  F ) )  /\  ( G R F )  =  ( J `  ( D `  F ) ) ) )
 
18.12.51  Functors
 
SyntaxcfuncOLD 25230 Extend class notation with the class of all functors.
 class  Func OLD
 
Syntaxcifunc 25231 Extend class notation with the class of all isomorphisms.
 class  Isofunc
 
Definitiondf-funcOLD 25232* 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.)
 |-  Func OLD 
 =  ( t  e. 
 Cat OLD  ,  u  e. 
 Cat OLD  |->  { f  e.  ( dom  ( dom_ `  u )  ^m  dom  ( dom_ `  t
 ) )  |  (
 A. o  e.  dom  ( id_ `  t ) E. p  e.  dom  ( id_ `  u )
 ( f `  (
 ( id_ `  t ) `  o ) )  =  ( ( id_ `  u ) `  p )  /\  ( A. m  e.  dom  ( dom_ `  t )
 ( f `  (
 ( id_ `  t ) `  ( ( dom_ `  t
 ) `  m )
 ) )  =  ( ( id_ `  u ) `  ( ( dom_ `  u ) `  (
 f `  m )
 ) )  /\  A. m  e.  dom  ( dom_ `  t ) ( f `
  ( ( id_ `  t ) `  (
 ( cod_ `  t ) `  m ) ) )  =  ( ( id_ `  u ) `  (
 ( cod_ `  u ) `  ( f `  m ) ) ) ) 
 /\  A. m  e.  dom  ( dom_ `  t ) A. n  e.  dom  ( dom_ `  t )
 ( ( ( cod_ `  t ) `  n )  =  ( ( dom_ `  t ) `  m )  ->  ( f `
  ( m ( o_ `  t ) n ) )  =  ( ( f `  m ) ( o_
 `  u ) ( f `  n ) ) ) ) }
 )
 
Theoremisfuna 25233* The class of functors between the categories  T and 
U. (Contributed by FL, 10-Feb-2008.)
 |-  O1  =  dom  ( id_ `  T )   &    |-  M1  =  dom  ( dom_ `  T )   &    |-  D1  =  ( dom_ `  T )   &    |-  C1  =  ( cod_ `  T )   &    |-  I1  =  ( id_ `  T )   &    |-  Ro 1  =  ( o_ `  T )   &    |-  O 2  =  dom  ( id_ `  U )   &    |-  M 2  =  dom  ( dom_ `  U )   &    |-  D 2  =  ( dom_ `  U )   &    |-  C 2  =  ( cod_ `  U )   &    |-  I 2  =  ( id_ `  U )   &    |-  Ro 2  =  ( o_ `  U )   =>    |-  ( ( T  e.  Cat
 OLD  /\  U  e.  Cat OLD  )  ->  ( Func OLD `  <. T ,  U >. )  =  { f  e.  ( M 2  ^m  M1 )  |  ( A. o  e.  O1  E. p  e.  O 2  ( f `
  ( I1 `  o
 ) )  =  ( I 2 `  p )  /\  ( A. m  e.  M1  ( f `  ( I1 `  ( D1 `  m ) ) )  =  ( I 2 `  ( D 2 `  ( f `  m ) ) )  /\  A. m  e.  M1  (
 f `  ( I1 `  ( C1
 `  m ) ) )  =  ( I 2 `  ( C 2 `  ( f `
  m ) ) ) )  /\  A. m  e.  M1  A. n  e.  M1  ( ( C1 `  n )  =  (
 D1 `  m )  ->  ( f `  ( m Ro 1 n ) )  =  ( ( f `  m ) Ro 2 ( f `
  n ) ) ) ) } )
 
Theoremisfunb 25234* The predicate "is a functor" . (Contributed by FL, 10-Feb-2008.)
 |-  O1  =  dom  ( id_ `  T )   &    |-  M1  =  dom  ( dom_ `  T )   &    |-  D1  =  ( dom_ `  T )   &    |-  C1  =  ( cod_ `  T )   &    |-  I1  =  ( id_ `  T )   &    |-  Ro 1  =  ( o_ `  T )   &    |-  O 2  =  dom  ( id_ `  U )   &    |-  M 2  =  dom  ( dom_ `  U )   &    |-  D 2  =  ( dom_ `  U )   &    |-  C 2  =  ( cod_ `  U )   &    |-  I 2  =  ( id_ `  U )   &    |-  Ro 2  =  ( o_ `  U )   =>    |-  ( ( T  e.  Cat
 OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  <->  ( F : M1 --> M 2  /\  ( A. o  e.  O1  E. p  e.  O 2  ( F `
  ( I1 `  o
 ) )  =  ( I 2 `  p )  /\  ( A. m  e.  M1  ( F `  ( I1 `  ( D1 `  m ) ) )  =  ( I 2 `  ( D 2 `  ( F `  m ) ) )  /\  A. m  e.  M1  ( F `
  ( I1 `  ( C1
 `  m ) ) )  =  ( I 2 `  ( C 2 `  ( F `
  m ) ) ) )  /\  A. m  e.  M1  A. n  e.  M1  ( ( C1 `  n )  =  (
 D1 `  m )  ->  ( F `  ( m Ro 1 n ) )  =  ( ( F `  m ) Ro 2 ( F `
  n ) ) ) ) ) ) )
 
Theoremfmamo 25235 A functor is a mapping between morphisms. (Contributed by FL, 10-Feb-2008.)
 |-  M1  =  dom  ( dom_ `  T )   &    |-  M 2  =  dom  ( dom_ `  U )   =>    |-  ( ( T  e.  Cat
 OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  ->  F : M1 --> M 2 )
 )
 
Theoremvidfunid 25236* The functor  F associates every object of  T to an object in  U. For the identification of objects with the identities see df-funcOLD 25232. JFM CAT1 th. 97. (Contributed by FL, 10-Feb-2008.)
 |-  O1  =  dom  ( id_ `  T )   &    |-  I1  =  ( id_ `  T )   &    |-  O 2  =  dom  ( id_ `  U )   &    |-  I 2  =  ( id_ `  U )   =>    |-  (
 ( T  e.  Cat OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  ->  A. o  e.  O1  E. p  e.  O 2  ( F `
  ( I1 `  o
 ) )  =  ( I 2 `  p ) ) )
 
Theoremiddvvidd 25237* Functors preserve domains. JFM CAT1 th. 98. (Contributed by FL, 5-May-2008.)
 |-  M1  =  dom  ( dom_ `  T )   &    |-  D1  =  ( dom_ `  T )   &    |-  I1  =  ( id_ `  T )   &    |-  I 2  =  ( id_ `  U )   &    |-  D 2  =  ( dom_ `  U )   =>    |-  (
 ( T  e.  Cat OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  ->  A. m  e.  M1  ( F `  ( I1 `  ( D1 `  m ) ) )  =  ( I 2 `  ( D 2 `  ( F `  m ) ) ) ) )
 
Theoremidcvvidc 25238* Functors preserve codomains. JFM CAT1 th. 98. (Contributed by FL, 5-May-2008.)
 |-  M1  =  dom  ( dom_ `  T )   &    |-  C1  =  ( cod_ `  T )   &    |-  I1  =  ( id_ `  T )   &    |-  I 2  =  ( id_ `  U )   &    |-  C 2  =  ( cod_ `  U )   =>    |-  (
 ( T  e.  Cat OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  ->  A. m  e.  M1  ( F `  ( I1 `  ( C1 `  m ) ) )  =  ( I 2 `  ( C 2 `  ( F `  m ) ) ) ) )
 
Theoremfmp 25239* Functors preserve morphisms composition. JFM CAT1 th. 99. (Contributed by FL, 2-Aug-2009.)
 |-  M1  =  dom  ( dom_ `  T )   &    |-  C1  =  ( cod_ `  T )   &    |-  D1  =  ( dom_ `  T )   &    |-  Ro 1  =  ( o_ `  T )   &    |-  Ro 2  =  ( o_ `  U )   =>    |-  ( ( T  e.  Cat
 OLD  /\  U  e.  Cat OLD  )  ->  ( F  e.  ( Func OLD `  <. T ,  U >. )  ->  A. m  e.  M1  A. n  e.  M1  ( ( C1 `  n )  =  ( D1 `  m )  ->  ( F `  ( m Ro 1 n ) )  =  ( ( F `  m ) Ro 2
 ( F `  n ) ) ) ) )
 
Theoremidfisf 25240 The identity functor is a functor. (Contributed by FL, 15-Jul-2008.)
 |-  ( T  e.  Cat OLD  ->  (  _I  |`  dom  ( dom_ `  T ) )  e.  ( Func OLD `  <. T ,  T >. ) )
 
Definitiondf-isof 25241* Class of isomorphisms. (Contributed by FL, 21-May-2012.)
 |-  Isofunc  =  ( u  e.  Cat OLD  ,  v  e.  Cat OLD  |->  { f  e.  ( Func OLD `  <. u ,  v >. )  |  E. g  e.  ( Func OLD `  <. v ,  u >. ) ( ( f  o.  g )  =  (  _I  |`  dom  ( dom_ `  v ) ) 
 /\  ( g  o.  f )  =  (  _I  |`  dom  ( dom_ `  u ) ) ) } )
 
18.12.52  Subcategories
 
Syntaxcsubcat 25242 Extend class notation with a function returning all the subcategories of a given category.
 class  SubCat
 
Definitiondf-subcat 25243  (  SubCat  `  x ) is the set of all the subcategories of the category  x. All the objects and all the morphisms of the subcategory belong to the supercategory. The identity of an object, the domain and the codomain of a morphism are the same in the subcategory and the supercategory. The composition of the subcategory is a restriction of the composition of the supercategory. (Contributed by FL, 17-Sep-2009.)
 |-  SubCat  =  ( x  e.  Cat OLD  |->  (  Cat OLD  i^i  ( ( ~P ( dom_ `  x )  X.  ~P ( cod_ `  x ) )  X.  ( ~P ( id_ `  x )  X.  ~P ( o_
 `  x ) ) ) ) )
 
Theoremissubcat 25244 The set of all the subcategories of 
T. (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   &    |-  J  =  ( id_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  (  SubCat  `  T )  =  (  Cat OLD 
 i^i  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R ) ) ) )
 
Theoremissubcata 25245 The predicate "is a subcategory of"  T. (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   &    |-  J  =  ( id_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  ( U  e.  (  SubCat  `  T )  <->  ( U  e.  Cat
 OLD  /\  ( ( id_ `  U )  C_  J  /\  ( ( dom_ `  U )  C_  D  /\  ( cod_ `  U )  C_  C )  /\  ( o_
 `  U )  C_  R ) ) ) )
 
Theoremissubcatb 25246 The predicate "is a subcategory of"  T. (Contributed by FL, 17-Sep-2009.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   &    |-  J  =  ( id_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  U  e.  Cat OLD  )  ->  ( U  e.  (  SubCat  `  T ) 
 <->  ( ( id_ `  U )  C_  J  /\  (
 ( dom_ `  U )  C_  D  /\  ( cod_ `  U )  C_  C )  /\  ( o_ `  U )  C_  R ) ) )
 
Theorembesubbeca 25247 Lemma to simplify some subcategories related theorems . (Contributed by FL, 17-Sep-2009.)
 |-  ( U  e.  (  SubCat  `  T )  ->  T  e.  Cat
 OLD  )
 
Theoremcatsbc 25248 A category belongs to the set of its subcategories. (Contributed by FL, 17-Sep-2009.)
 |-  ( T  e.  Cat OLD  ->  T  e.  (  SubCat  `  T ) )
 
Theoremobsubc2 25249 The objects of a subcategory are a subset of those of the supercategory. JFM CAT2 th. 11 . (Contributed by FL, 17-Sep-2009.)
 |-  O1  =  dom  ( id_ `  T )   &    |-  O 2  =  dom  ( id_ `  U )   =>    |-  ( U  e.  (  SubCat  `  T )  ->  O 2  C_  O1 )
 
Theoremidsubc 25250 The identity function of a subcategory is a subset of the identity function of the supercategory. (Contributed by FL, 17-Sep-2009.)
 |-  I1  =  ( id_ `  T )   &    |-  I 2  =  ( id_ `  U )   =>    |-  ( U  e.  (  SubCat  `  T )  ->  I 2  C_  I1 )
 
Theoremdomsubc 25251 The domain function of a subcategory is a subset of the domain function of the supercategory. (Contributed by FL, 19-Sep-2009.)
 |-  D1  =  ( dom_ `  T )   &    |-  D 2  =  ( dom_ `  U )   =>    |-  ( U  e.  (  SubCat  `  T )  ->  D 2  C_  D1 )
 
Theoremcodsubc 25252 The codomain function of a subcategory is a subset of the codomain function of the supercategory. (Contributed by FL, 19-Sep-2009.)
 |-  C1  =  ( cod_ `  T )   &    |-  C 2  =  ( cod_ `  U )   =>    |-  ( U  e.  (  SubCat  `  T )  ->  C 2  C_  C1 )
 
Theoremsubctct 25253 A subcategory is a category. (Contributed by FL, 17-Sep-2009.)
 |-  ( U  e.  (  SubCat  `  T )  ->  U  e.  Cat
 OLD  )
 
Theoremmorsubc 25254 The morphisms of a subcategory are a subset of those of the supercategory. (Contributed by FL, 18-Sep-2009.)
 |-  M1  =  dom  ( dom_ `  T )   &    |-  M 2  =  dom  ( dom_ `  U )   =>    |-  ( U  e.  (  SubCat  `  T )  ->  M 2  C_  M1 )
 
Theoremcmpsubc 25255 The composition law of a subcategory is a subset of the composition law of the supercategory. (Contributed by FL, 20-Sep-2009.)
 |-  Ro 1  =  ( o_ `  T )   &    |-  Ro 2  =  ( o_ `  U )   =>    |-  ( U  e.  (  SubCat  `  T )  ->  Ro 2  C_ 
 Ro 1 )
 
Theoremidsubidsup 25256* The identity of an an objet of the subcategory equals the identity of the object in the supercategory. (Contributed by FL, 2-Nov-2009.)
 |-  I1  =  ( id_ `  T )   &    |-  I 2  =  ( id_ `  U )   &    |-  O 2  =  dom  ( id_ `  U )   =>    |-  ( U  e.  (  SubCat  `  T )  ->  A. x  e.  O 2  ( I 2 `  x )  =  ( I1 `  x ) )
 
Theoremidsubfun 25257 The identity restricted to the morphism of a subcategory  U is a functor from the subcategory to the supercategory. It is called the inclusion functor. JFM CAT2 th. 19. (Contributed by FL, 5-Oct-2009.)
 |-  M  =  dom  ( dom_ `  U )   =>    |-  ( U  e.  (  SubCat  `  T )  ->  (  _I  |`  M )  e.  ( Func OLD `  <. U ,  T >. ) )
 
Theoreminfemb 25258 The inclusion functor is an embedding. (Contributed by FL, 2-Nov-2009.)
 |-  M1  =  dom  ( dom_ `  T )   &    |-  M 2  =  dom  ( dom_ `  U )   =>    |-  ( U  e.  (  SubCat  `  T )  ->  (  _I  |`  M 2 ) : M 2 -1-1-> M1 )
 
18.12.53  Terminal and initial objects
 
Syntaxciobj 25259 Extend class notation with the class of all initial objects.
 class  InitObj
 
Definitiondf-inob 25260* Definition of the initial objects of a category. Experimental. (Contributed by FL, 27-Jun-2014.)
 |-  InitObj  =  ( x  e.  Cat OLD  |->  { z  e.  dom  ( id_ `  x )  | 
 A. o  e.  dom  ( id_ `  x ) E! m  e.  dom  ( dom_ `  x )
 ( ( ( dom_ `  x ) `  m )  =  z  /\  ( ( cod_ `  x ) `  m )  =  o ) } )
 
Theoremisinob 25261* The predicate "are the initial objects of a category". (Contributed by FL, 27-Jun-2014.)
 |-  O  =  dom  ( id_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Cat OLD  ->  ( 
 InitObj  `  T )  =  { z  e.  O  |  A. o  e.  O  E! m  e.  M  ( ( D `  m )  =  z  /\  ( C `  m )  =  o ) } )
 
Syntaxctobj 25262 Extend class notation with the class of all terminal objects.
 class  TermObj
 
Definitiondf-termob 25263* Definition of the terminal objects of a category. Experimental. (Contributed by FL, 15-Jul-2012.)
 |-  TermObj  =  ( x  e.  Cat OLD  |->  { z  e.  dom  ( id_ `  x )  | 
 A. o  e.  dom  ( id_ `  x ) E! m  e.  dom  ( dom_ `  x )
 ( ( ( dom_ `  x ) `  m )  =  o  /\  ( ( cod_ `  x ) `  m )  =  z ) } )
 
18.12.54  Sources and sinks
 
Syntaxcsrce 25264 Extend class notation with the class of all sources.
 class  Source
 
Definitiondf-source 25265* A source is a family  s of morphims indexed by a set  i which all have the same domain  d. Joy of Cats, def. 10.1, p. 169. Experimental. (Contributed by FL, 30-May-2014.)
 |-  Source  =  ( c  e.  Cat OLD  ,  i  e.  _V  |->  { s  e.  ( dom  ( dom_ `  c )  ^m  i )  |  A. x  e.  i  A. y  e.  i  (
 ( dom_ `  c ) `  ( s `  x ) )  =  (
 ( dom_ `  c ) `  ( s `  y
 ) ) } )
 
Theoremissrc 25266* Properties of a source. (Contributed by FL, 27-Jun-2014.)
 |-  D  =  ( dom_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  I  e.  A )  ->  ( S  e.  ( T  Source  I )  <-> 
 ( S  e.  ( M  ^m  I )  /\  A. x  e.  I  A. y  e.  I  ( D `  ( S `  x ) )  =  ( D `  ( S `  y ) ) ) ) )
 
Theorempropsrc 25267* Properties of a source. (Contributed by FL, 30-May-2014.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  S  =  ( T  Source  I )   =>    |-  ( ( T  e.  Cat OLD  /\  I  e.  A  /\  F  e.  S )  ->  ( F : I --> M  /\  A. x  e.  I  A. y  e.  I  ( D `  ( F `  x ) )  =  ( D `
  ( F `  y ) ) ) )
 
Syntaxcsnk 25268 Extend class notation with the class of all sinks.
 class  Sink
 
Definitiondf-sink 25269* A sink is a family of morphims indexed by a set  i which all have the same codomain. Joy of Cats, def. 10.62, p. 184. Experimental. (Contributed by FL, 15-Jul-2012.)
 |-  Sink  =  ( c  e.  Cat OLD 
 ,  i  e.  _V  |->  { s  e.  ( dom  ( dom_ `  c )  ^m  i )  |  A. x  e.  i  A. y  e.  i  (
 ( cod_ `  c ) `  ( s `  x ) )  =  (
 ( cod_ `  c ) `  ( s `  y
 ) ) } )
 
Syntaxcntrl 25270 Extend class notation with the class of all natural sources.
 class  Natural
 
Definitiondf-natur 25271* A diagram is an indexed family of objects and morphisms in a category C. Maybe more than one morphism between two given objects, maybe none. One might choose a simple set with no structure for the set of indices as usual but we can also use a category. Let's call I this category. Morphisms of I are used as indices of morphisms of C and objects of I are used as indices of objects of C. With this convention a diagram is now a functor  e.  ( Func OLD `  <. I ,  C >. ).

Using a category for the indices is even a better solution than using a simple set because with the functions dom_ and cod_ it is easy to express the relationship between a morphism in C and its attached objects simply by naming the relationship between the morphism in I and its attached objects (let's recall a functor preserves the domain and codomain).

So ...

Let  d  e.  (
Func OLD `  <. i ,  c >. ) be a diagram, a source indexed by the objects of  i is said to be natural for the diagram provided that each morphism  m of  d and the morphisms of the source connected to its domain and codomain commute. Joy of Cats, def. 11.3, (1) p. 193. Goldblatt calls "cone" what Adamek, Herrlich and Strecker call "natural source". Experimental. (Contributed by FL, 27-Jun-2014.)

 |-  Natural  =  ( i  e.  Cat OLD  ,  t  e.  Cat OLD  |->  ( d  e.  ( Func OLD `  <. i ,  t >. )  |->  { s  e.  ( t  Source  dom  ( id_ `  i ) )  |  A. m  e. 
 dom  ( dom_ `  i
 ) ( ( (
 dom_ `  t ) `  ( d `  m ) )  =  (
 ( cod_ `  t ) `  ( s `  (
 ( dom_ `  i ) `  m ) ) ) 
 /\  ( ( cod_ `  t ) `  (
 d `  m )
 )  =  ( (
 cod_ `  t ) `  ( s `  (
 ( cod_ `  i ) `  m ) ) ) 
 /\  ( ( d `
  m ) ( o_ `  t ) ( s `  (
 ( dom_ `  i ) `  m ) ) )  =  ( s `  ( ( cod_ `  i
 ) `  m )
 ) ) } )
 )
 
Theoremisntr 25272* The predicate "is a natural source". (Contributed by FL, 27-Jun-2014.)
 |-  O1  =  dom  ( id_ `  I
 )   &    |-  M1  =  dom  ( dom_ `  I )   &    |-  D1  =  ( dom_ `  I )   &    |-  C1  =  ( cod_ `  I )   &    |-  D 2  =  ( dom_ `  T )   &    |-  C 2  =  ( cod_ `  T )   &    |-  Ro 2  =  ( o_ `  T )   =>    |-  ( ( I  e. 
 Cat OLD  /\  T  e.  Cat
 OLD  /\  D  e.  ( Func OLD `  <. I ,  T >. ) )  ->  ( S  e.  (
 ( I  Natural  T ) `
  D )  <->  ( S  e.  ( T  Source  O1 )  /\  A. m  e.  M1  ( ( D 2 `  ( D `  m ) )  =  ( C 2 `  ( S `
  ( D1 `  m ) ) )  /\  ( C 2 `  ( D `  m ) )  =  ( C 2 `  ( S `  ( C1
 `  m ) ) )  /\  ( ( D `  m ) Ro 2 ( S `
  ( D1 `  m ) ) )  =  ( S `  ( C1
 `  m ) ) ) ) ) )
 
18.12.55  Limits and co-limits
 
Syntaxclmct 25273 Extend class notation with the class of all limits.
 class  LimCat
 
Definitiondf-limcat 25274* A limit of a diagram  d is a natural source  l for the diagram with the universal property that every natural source for  d uniquely factors through it. Joy of Cats, def. 11.3 (2), p. 194. Experimental. (Contributed by FL, 27-Jun-2014.)
 |-  LimCat  =  ( i  e.  Cat OLD  ,  t  e.  Cat OLD  |->  ( d  e.  ( Func OLD `  <. i ,  t >. )  |->  { l  e.  ( ( i  Natural  t ) `  d )  |  A. f  e.  ( ( i  Natural  t ) `  d ) E! m  e.  dom  ( dom_ `  t ) A. x  e.  dom  ( id_ `  i )
 ( f `  x )  =  ( (
 l `  x )
 ( o_ `  t
 ) m ) }
 ) )
 
Theoremislimcat 25275* The predicate "is a limit of a diagram." (Contributed by FL, 27-Jun-2014.)
 |-  O1  =  dom  ( id_ `  I
 )   &    |-  M 2  =  dom  ( dom_ `  T )   &    |-  Ro 2  =  ( o_ `  T )   =>    |-  ( ( I  e. 
 Cat OLD  /\  T  e.  Cat
 OLD  /\  D  e.  ( Func OLD `  <. I ,  T >. ) )  ->  ( L  e.  (
 ( I  LimCat  T ) `
  D )  <->  ( L  e.  ( ( I  Natural  T ) `  D ) 
 /\  A. f  e.  (
 ( I  Natural  T ) `
  D ) E! m  e.  M 2  A. x  e.  O1  (
 f `  x )  =  ( ( L `  x ) Ro 2 m ) ) ) )
 
18.12.56  Product and sum of two objects
 
Syntaxcprodo 25276 Extend class notation with the class of all object products.
 class  Prod Obj
 
Definitiondf-prodobj 25277* " A product in a category  C of two objects  a and  b is a  C-object  a  x.  b together with a pair ( m : a  x.  b --> a,  n : a  x.  b --> b) of  C-arrows such that for any pair of  C -arrows of the form ( f : c --> a,  g : c --> b) there is exactly one arrow  <. f ,  g
>. : c --> a  x.  b such that  m  o.  <. f ,  g >.  =  f and  n  o.  <. f ,  g >.  =  g". Goldblatt p. 47. Experimental. (Contributed by FL, 15-Jul-2012.)
 |-  Prod Obj 
 =  ( c  e. 
 Cat OLD  |->  ( a  e. 
 dom  ( id_ `  c
 ) ,  b  e. 
 dom  ( id_ `  c
 )  |->  { <. m ,  n >.  |  ( ( m  e.  dom  ( dom_ `  c )  /\  n  e.  dom  ( dom_ `  c
 ) )  /\  (
 ( ( dom_ `  c
 ) `  m )  =  ( ( dom_ `  c
 ) `  n )  /\  ( ( cod_ `  c
 ) `  m )  =  a  /\  ( (
 cod_ `  c ) `  n )  =  b
 )  /\  A. f  e. 
 dom  ( dom_ `  c
 ) A. g  e.  dom  ( dom_ `  c )
 ( ( ( (
 dom_ `  c ) `  f )  =  (
 ( dom_ `  c ) `  g )  /\  (
 ( cod_ `  c ) `  f )  =  a 
 /\  ( ( cod_ `  c ) `  g
 )  =  b ) 
 /\  E! h  e.  dom  ( dom_ `  c )
 ( ( ( dom_ `  c ) `  h )  =  ( ( dom_ `  c ) `  f )  /\  ( (
 cod_ `  c ) `  h )  =  (
 ( dom_ `  c ) `  m )  /\  (
 ( m ( o_
 `  c ) h )  =  f  /\  ( n ( o_ `  c ) h )  =  g ) ) ) ) } )
 )
 
Syntaxcsumo 25278 Extend class notation with the class of all object sums.
 class  Sum Obj
 
Definitiondf-sumobj 25279* " A co-product of C-objects  a and  b is a C-object  a  +  b together with a pair ( m : a --> a  +  b,  n : b --> a  +  b) of C-arrows such that for any pair of C-arrows of the form ( f : a --> c,  g : b --> c) there is exactly one arrow  [ f ,  g ] : a  +  b --> c such that  [ f ,  g ]  o.  m  =  f and  [ f ,  g ]  o.  n  =  g". Goldblatt p. 54. Experimental. (Contributed by FL, 15-Jul-2012.)
 |-  Sum Obj 
 =  ( c  e. 
 Cat OLD  |->  ( a  e. 
 dom  ( id_ `  c
 ) ,  b  e. 
 dom  ( id_ `  c
 )  |->  { <. m ,  n >.  |  ( ( m  e.  dom  ( dom_ `  c )  /\  n  e.  dom  ( dom_ `  c
 ) )  /\  (
 ( ( cod_ `  c
 ) `  m )  =  ( ( cod_ `  c
 ) `  n )  /\  ( ( dom_ `  c
 ) `  m )  =  a  /\  ( (
 dom_ `  c ) `  n )  =  b
 )  /\  A. f  e. 
 dom  ( dom_ `  c
 ) A. g  e.  dom  ( dom_ `  c )
 ( ( ( (
 cod_ `  c ) `  f )  =  (
 ( cod_ `  c ) `  g )  /\  (
 ( dom_ `  c ) `  f )  =  a 
 /\  ( ( dom_ `  c ) `  g
 )  =  b ) 
 /\  E! h  e.  dom  ( dom_ `  c )
 ( ( ( cod_ `  c ) `  h )  =  ( ( cod_ `  c ) `  f )  /\  ( (
 dom_ `  c ) `  h )  =  (
 ( cod_ `  c ) `  m )  /\  (
 ( h ( o_
 `  c ) m )  =  f  /\  ( h ( o_ `  c ) n )  =  g ) ) ) ) } )
 )
 
18.12.57  Tarski's classes
 
Syntaxctar 25280 Extends class notation to include function  tar.
 class  tar
 
Definitiondf-tar 25281* A function to study Tarski's classes. See valdom 25283 for its domain, vtare 25284 for its value at  (/), vtarsu 25285 for its value at a successor, vtarl 25286 for its value at a limit ordinal. (Contributed by FL, 20-Mar-2011.)
 |-  tar  =  ( a  e.  _V ,  b  e.  _V  |->  ( rec ( ( x  e.  _V  |->  ( ( { u  |  E. v  e.  x  ( u  C_  v  \/  u  =  ~P v ) }  u.  ~P x )  i^i  ( tarskiMap `  a )
 ) ) ,  {
 a } )  |`  b ) )
 
Theoremvaltar 25282* The  tar function as a recursive function. (Contributed by FL, 20-Mar-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  (
 ( X  e.  A  /\  Y  e.  B ) 
 ->  ( tar `  <. X ,  Y >. )  =  ( rec ( ( x  e.  _V  |->  ( ( { u  |  E. v  e.  x  ( u  C_  v  \/  u  =  ~P v ) }  u.  ~P x )  i^i  ( tarskiMap `  X )
 ) ) ,  { X } )  |`  Y ) )
 
Theoremvaldom 25283 The domain of the  tar function is the ordinal  Y. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( X  e.  A  /\  Y  e.  On )  ->  dom  ( tar `  <. X ,  Y >. )  =  Y )
 
Theoremvtare 25284 Value of  tar at  (/). JFM CLASSES1 th. 10. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( X  e.  A  /\  Y  e.  On  /\  Y  =/=  (/) )  ->  (
 ( tar `  <. X ,  Y >. ) `  (/) )  =  { X } )
 
Theoremvtarsu 25285* The parts and the powersets of the elements of  tar ( A ) are elements of  tar ( suc  A
). As well as the parts of  tar ( A ) when they are elements of the smallest Tarski's class of which  X is an element. JFM CLASSES1 th. 11. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( X  e.  B  /\  Y  e.  On  /\  suc 
 A  e.  Y ) 
 ->  ( ( tar `  <. X ,  Y >. ) `  suc  A )  =  ( ( { u  |  E. v  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u  C_  v  \/  u  =  ~P v ) }  u.  ~P ( ( tar `  <. X ,  Y >. ) `  A ) )  i^i  ( tarskiMap `  X ) ) )
 
Theoremvtarl 25286 The value of  tar at a limit ordinal. JFM CLASSES1 th. 12. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( X  e.  B  /\  Y  e.  On  /\  ( A  e.  Y  /\  Lim  A ) ) 
 ->  ( ( tar `  <. X ,  Y >. ) `  A )  =  U. ( ( tar `  <. X ,  Y >. ) " A ) )
 
Theoremtartarmap 25287 The sequence  tar has its values in a Tarski's class. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( X  e.  A  /\  Y  e.  On  /\  suc 
 B  e.  Y ) 
 ->  ( ( tar `  <. X ,  Y >. ) `  B )  C_  ( tarskiMap `  X )
 )
 
Theorempwtsm 25288 If  A belongs to the smallest Tarski's class that contains  X so does  ~P A. CLASSES1 th. 7. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
 |-  ( A  e.  ( tarskiMap `  X )  ->  ~P A  e.  ( tarskiMap `  X ) )
 
Theoremsubtsm 25289 If  A belongs to the smallest Tarski's class that contains  X so does the subsets of  A. CLASSES1. th. 6. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
 |-  ( A  e.  ( tarskiMap `  X )  ->  ~P A  C_  ( tarskiMap `  X ) )
 
Theoremsubtareqbe 25290 If  A is a subset of the smallest Tarski's class that contains  X then it is equipotent to this class or it belongs to it. CLASSES1 th. 8. (Contributed by FL, 17-Apr-2011.)
 |-  ( A  C_  ( tarskiMap `  X )  ->  ( A  ~~  ( tarskiMap `  X )  \/  A  e.  ( tarskiMap `  X )
 ) )
 
Syntaxctr 25291 Extend class notation to include the function whose value is the transitive closure of its operand.
 class  tr
 
Definitiondf-trcls 25292* The transitive closure of a set. (Contributed by FL, 17-Apr-2011.)
 |-  tr  =  ( x  e.  _V  |->  U_ a  e.  om  (
 ( rec ( ( z  e.  _V  |->  ( z  u.  U. z ) ) ,  x )  |`  om ) `  a
 ) )
 
Theoremtrclval 25293* The transitive closure of a set A. (Contributed by FL, 17-Apr-2011.)
 |-  ( A  e.  B  ->  ( tr `  A )  =  U_ a  e. 
 om  ( ( rec ( ( z  e. 
 _V  |->  ( z  u. 
 U. z ) ) ,  A )  |`  om ) `  a ) )
 
Theoremvtarsuelt 25294* C belongs to the value of  tar at a successor of  A iff it is a part of  tar at  A, the powerset of an element or a part of an element of  tar at  A. CLASSES1 th. 13 (Contributed by FL, 13-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
 |-  (
 ( X  e.  B  /\  Y  e.  On  /\  suc 
 A  e.  Y ) 
 ->  ( C  e.  (
 ( tar `  <. X ,  Y >. ) `  suc  A )  <->  ( ( C 
 C_  ( ( tar `  <. X ,  Y >. ) `  A ) 
 /\  C  e.  ( tarskiMap `  X ) )  \/ 
 E. z  e.  (
 ( tar `  <. X ,  Y >. ) `  A ) ( C  C_  z  \/  C  =  ~P z ) ) ) )
 
Theorempartarelt1 25295 If  C is a part of an element of our tar function at  A then  C is an element or tar at 
suc  A. CLASSES1 th. 14 (Contributed by FL, 13-Apr-2011.)
 |-  (
 ( X  e.  B  /\  Y  e.  On  /\  suc 
 A  e.  Y ) 
 ->  ( ( C  C_  Z  /\  Z  e.  (
 ( tar `  <. X ,  Y >. ) `  A ) )  ->  C  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A ) ) )
 
Theorempartarelt2 25296 If  Z is an element of our tar function at  A then  ~P Z is an element or tar at  suc  A. CLASSES1 th. 15 (Contributed by FL, 13-Apr-2011.)
 |-  (
 ( X  e.  B  /\  Y  e.  On  /\  suc 
 A  e.  Y ) 
 ->  ( Z  e.  (
 ( tar `  <. X ,  Y >. ) `  A )  ->  ~P Z  e.  (
 ( tar `  <. X ,  Y >. ) `  suc  A ) ) )
 
Theoremtareltsuc 25297 All the element of  tar at  A are elements of  tar at  suc  A. CLASSES1 th. 18 (Contributed by FL, 13-Apr-2011.)
 |-  (
 ( X  e.  B  /\  Y  e.  On  /\  suc 
 A  e.  Y ) 
 ->  ( ( tar `  <. X ,  Y >. ) `  A )  C_  ( ( tar `  <. X ,  Y >. ) `  suc  A ) )
 
Theoremeltintpar 25298 An element of the intersection of a Tarski's class with the class of the ordinal numbers is a part of the intersection. (Contributed by FL, 20-Apr-2011.)
 |-  ( T  e.  Tarski  ->  ( A  e.  ( On  i^i  T )  ->  A  C_  ( On  i^i  T ) ) )
 
Theoreminttaror 25299 The intersection of a Tarski's class with the class of the ordinal numbers is an ordinal number. (Contributed by FL, 20-Apr-2011.)
 |-  ( T  e.  Tarski  ->  ( On  i^i  T )  e. 
 On )
 
Theoreminttarcar 25300 The intersection of a Tarski's class and the ordinal numbers is equipotent to the Tarski's class. JFM CLASSES2. . (Contributed by FL, 20-Apr-2011.)
 |-  ( T  e.  Tarski  ->  ( On  i^i  T )  ~~  T )
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