HomeHome Metamath Proof Explorer
Theorem List (p. 252 of 314)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21444)
  Hilbert Space Explorer  Hilbert Space Explorer
(21445-22967)
  Users' Mathboxes  Users' Mathboxes
(22968-31305)
 

Theorem List for Metamath Proof Explorer - 25101-25200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrdmob 25101 The range of  ( dom_ `  T
) is the class of the objects. (Contributed by FL, 10-Jan-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  D  =  ( dom_ `  T )   =>    |-  ( T  e.  Ded  ->  ran  D  =  O )
 
Theoremrcmob 25102 The range of  ( cod_ `  T
) is the class of the objects. (Contributed by FL, 10-Jan-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Ded  ->  ran  C  =  O )
 
Theoremaidm2 25103 The underlying directed multi graph of a deductive system. (Contributed by FL, 20-Sep-2009.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Ded  ->  <. <. D ,  C >. ,  ran  D >.  e.  Dgra )
 
Theoremdmrngcmp 25104 Domain and range of the domain of the composition. (Contributed by FL, 5-Oct-2009.)
 |-  R  =  ( o_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   =>    |-  ( T  e.  Ded 
 ->  ( dom  dom  R  =  M  /\  ran  dom  R  =  M ) )
 
16.12.48  Categories
 
SyntaxccatOLD 25105 Extend class notation with the class of categories.
 class  Cat OLD
 
Definitiondf-catOLD 25106* A category is a deductive system where composition is associative, and identities are neutral elements. A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated to those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.)
 |-  Cat OLD 
 =  { x  |  E. d E. c E. j E. r ( x  =  <. <. d ,  c >. ,  <. j ,  r >.
 >.  /\  ( ( <. <.
 d ,  c >. , 
 <. j ,  r >. >.  e.  Ded  /\  A. f  e. 
 dom  d A. g  e.  dom  d A. h  e.  dom  d ( ( ( d `  h )  =  ( c `  g )  /\  (
 d `  g )  =  ( c `  f
 ) )  ->  ( h r ( g r f ) )  =  ( ( h r g ) r f ) ) ) 
 /\  ( A. a  e.  dom  j A. f  e.  dom  d ( ( c `  f )  =  a  ->  (
 ( j `  a
 ) r f )  =  f )  /\  A. a  e.  dom  j A. f  e.  dom  d ( ( d `
  f )  =  a  ->  ( f
 r ( j `  a ) )  =  f ) ) ) ) }
 
TheoremiscatOLD 25107* The predicate "is a category". (Contributed by FL, 24-Oct-2007.)
 |-  M  =  dom  D   &    |-  O  =  dom  J   =>    |-  ( ( ( D  e.  A  /\  C  e.  B  /\  J  e.  F )  /\  R  e.  G )  ->  ( <. <. D ,  C >. , 
 <. J ,  R >. >.  e.  Cat OLD  <->  ( ( <. <. D ,  C >. , 
 <. J ,  R >. >.  e.  Ded  /\  A. f  e.  M  A. g  e.  M  A. h  e.  M  ( ( ( D `  h )  =  ( C `  g )  /\  ( D `
  g )  =  ( C `  f
 ) )  ->  ( h R ( g R f ) )  =  ( ( h R g ) R f ) ) )  /\  ( A. a  e.  O  A. f  e.  M  ( ( C `  f
 )  =  a  ->  ( ( J `  a ) R f )  =  f ) 
 /\  A. a  e.  O  A. f  e.  M  ( ( D `  f
 )  =  a  ->  ( f R ( J `  a ) )  =  f ) ) ) ) )
 
Theoremcati 25108* Definitional properties of a category. (Contributed by FL, 24-Oct-2007.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_
 `  T )   &    |-  M  =  dom  D   &    |-  O  =  dom  J   =>    |-  ( T  e.  Cat OLD 
 ->  ( ( <. <. D ,  C >. ,  <. J ,  R >. >.  e.  Ded  /\  A. f  e.  M  A. g  e.  M  A. h  e.  M  ( ( ( D `  h )  =  ( C `  g )  /\  ( D `
  g )  =  ( C `  f
 ) )  ->  ( h R ( g R f ) )  =  ( ( h R g ) R f ) ) )  /\  ( A. a  e.  O  A. f  e.  M  ( ( C `  f
 )  =  a  ->  ( ( J `  a ) R f )  =  f ) 
 /\  A. a  e.  O  A. f  e.  M  ( ( D `  f
 )  =  a  ->  ( f R ( J `  a ) )  =  f ) ) ) )
 
Theorem0alg 25109 Lemma for 0ded 25110. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Alg
 
Theorem0ded 25110 A deductive system with no object and no morphism. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Ded
 
Theorem0catOLD 25111 Category  0 has no object and no morphism. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Cat OLD
 
Theorem1cat 25112 Category  1 has one object and one morphism. (Contributed by FL, 30-Oct-2007.)
 |-  A  e.  _V   =>    |- 
 <. <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
 >. ,  <. { <. A ,  <. A ,  A >. >. } ,  { <. <. <. A ,  A >. ,  <. A ,  A >. >. ,  <. A ,  A >. >. } >. >.  e.  Cat OLD
 
Theoremstrcat 25113 Structure of a category. (Contributed by FL, 26-Oct-2007.)
 |-  Cat OLD  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
 
Theoremrelcat 25114 A category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  Cat
 OLD
 
Theoremreldcat 25115 The domain of a category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  dom 
 Cat OLD
 
Theoremrelrcat 25116 The range of a category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  ran 
 Cat OLD
 
Theoremcatded 25117 A category is a deductive system. (Contributed by FL, 26-Oct-2007.)
 |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
 
Theoremdomc 25118 The 1st "axiom" of a category:  ( dom_ `  T ) is a mapping from the morphisms of  T to the objects of  T. (Contributed by FL, 2-Jan-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   =>    |-  ( T  e.  Cat OLD  ->  D : M --> O )
 
Theoremcodc 25119 The 2nd "axiom" of a category  ( cod_ `  T ) is a mapping from the morphisms of  T to the objects of  T. (Contributed by FL, 2-Jan-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Cat OLD  ->  C : M --> O )
 
Theoremidc 25120 The 3rd "axiom" of a category  ( id_ `  T ) is a mapping from the objects of  T to the morphisms of  T. (Contributed by FL, 5-Dec-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   =>    |-  ( T  e.  Cat OLD  ->  J : O --> M )
 
Theoremcmppfc 25121 The 4th "axiom" of a category:  ( o_ `  T ) is a partial operation from the morphisms of  T to the morphisms of  T. (Contributed by FL, 10-Mar-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  ( Fun  R  /\  dom 
 R  C_  ( M  X.  M )  /\  ran  R 
 C_  M ) )
 
Theoremidosc 25122 The 5th "axiom" of a category: identities are morphisms whose domains and codomains are equal. (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  J   &    |-  D  =  (
 dom_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O )  ->  ( ( D `
  ( J `  A ) )  =  A  /\  ( C `
  ( J `  A ) )  =  A ) )
 
Theoremcmppfcd 25123 The 6th "axiom" of a category:  ( G ( o_ `  T ) F ) is only defined when the domain of  F equals the codomain of 
G. (Contributed by FL, 10-Mar-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  F  e.  M  /\  G  e.  M ) 
 ->  ( <. G ,  F >.  e.  dom  R  <->  ( D `  G )  =  ( C `  F ) ) )
 
Theoremdomcmpc 25124 The 7th "axiom" of a category: when  ( G ( o_ `  T ) F ) is defined its domain is the domain of 
F. (Contributed by FL, 10-Mar-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  F  e.  M  /\  G  e.  M ) 
 ->  ( ( D `  G )  =  ( C `  F )  ->  ( D `  ( G R F ) )  =  ( D `  F ) ) )
 
Theoremcodcmpc 25125 The 8th "axiom" of a category: when  ( G ( o_ `  T ) F ) is defined its codomain is the codomain of  G. (Contributed by FL, 10-Jan-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  F  e.  M  /\  G  e.  M ) 
 ->  ( ( D `  G )  =  ( C `  F )  ->  ( C `  ( G R F ) )  =  ( C `  G ) ) )
 
Theoremcmpasso 25126 The 9th "axiom" of a category:  ( o_ `  T ) is associative. (Contributed by FL, 29-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( F  e.  M  /\  G  e.  M  /\  H  e.  M ) )  ->  ( (
 ( D `  H )  =  ( C `  G )  /\  ( D `  G )  =  ( C `  F ) )  ->  ( H R ( G R F ) )  =  ( ( H R G ) R F ) ) )
 
Theoremcmpida 25127 The 10th "axiom" of a category:  ( J `  A ) is a left neutral element. (Contributed by FL, 29-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O  /\  F  e.  M ) 
 ->  ( ( C `  F )  =  A  ->  ( ( J `  A ) R F )  =  F )
 )
 
Theoremcmpidb 25128 The 11th "axiom" of a category:  ( J `  A ) is a right neutral element. (Contributed by FL, 24-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O  /\  F  e.  M ) 
 ->  ( ( D `  F )  =  A  ->  ( F R ( J `  A ) )  =  F ) )
 
Theoremdmo 25129 The domain of a morphism is an object. (Contributed by FL, 2-Jan-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  O  =  dom  ( id_ `  T )   &    |-  D  =  ( dom_ `  T )   =>    |-  (
 ( T  e.  Cat OLD  /\  F  e.  M ) 
 ->  ( D `  F )  e.  O )
 
Theoremcdmo 25130 The codomain of a morphism is an object. (Contributed by FL, 2-Jan-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  O  =  dom  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  (
 ( T  e.  Cat OLD  /\  F  e.  M ) 
 ->  ( C `  F )  e.  O )
 
Theoremjdmo 25131 An identity is a morphism. (Contributed by FL, 10-Jan-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  O  =  dom  ( id_ `  T )   &    |-  J  =  ( id_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O )  ->  ( J `  A )  e.  M )
 
Theoremcmpmorp 25132 Conditions for a composite to be a morphism. (Contributed by FL, 10-Mar-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  F  e.  M  /\  G  e.  M ) 
 ->  ( ( D `  G )  =  ( C `  F )  ->  ( G R F )  e.  M ) )
 
Theoremmorcat 25133 Two ways to define the set of the morphisms of a category. (Contributed by FL, 19-Sep-2009.)
 |-  ( T  e.  Cat OLD  ->  dom  ( dom_ `  T )  =  dom  ( cod_ `  T ) )
 
Theoremcmppfc1 25134 Composition is a function. (Contributed by FL, 5-Oct-2009.)
 |-  R  =  ( o_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  Fun  R )
 
Theoremdualalg 25135 The dual of a  Alg is a  Alg. (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_
 `  T )   =>    |-  ( T  e.  Alg 
 ->  <. <. C ,  D >. ,  <. J , tpos  R >.
 >.  e.  Alg  )
 
Theoremdualded 25136 The dual of a deductive system is a deductive system. (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_
 `  T )   =>    |-  ( T  e.  Ded 
 ->  <. <. C ,  D >. ,  <. J , tpos  R >.
 >.  e.  Ded )
 
Theoremdualcat2 25137 The dual of a category is a category. Joy of cats 3.5 (Contributed by FL, 4-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_
 `  T )   =>    |-  ( T  e.  Cat
 OLD  ->  <. <. C ,  D >. ,  <. J , tpos  R >.
 >.  e.  Cat OLD  )
 
16.12.49  Homsets
 
SyntaxchomOLD 25138 Extend class notation with the function returning all the morphisms between two objects.
 class  hom
 
Definitiondf-homOLD 25139*  ( hom `  x ) is a function which returns for each pair of objects  <. a ,  b >. the morphisms whose domain is  a and codomain  b. JFM CAT1 def. 6 (Contributed by FL, 6-May-2007.)
 |-  hom  =  ( x  e.  Cat OLD  |->  ( a  e.  dom  ( id_ `  x ) ,  b  e.  dom  ( id_ `  x )  |->  { f  e.  dom  ( dom_ `  x )  |  ( ( ( dom_ `  x ) `  f
 )  =  a  /\  ( ( cod_ `  x ) `  f )  =  b ) } )
 )
 
Theoremishoma 25140* Definition of  ( hom `  T
). (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  O  =  dom  ( id_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Cat OLD  ->  ( hom `  T )  =  ( a  e.  O ,  b  e.  O  |->  { f  e.  M  |  ( ( D `  f )  =  a  /\  ( C `  f
 )  =  b ) } ) )
 
Theoremishomb 25141* The homset  ( ( hom `  T ) `  <. A ,  B >. ). (Contributed by FL, 18-May-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  T  e.  Cat OLD   =>    |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( H `  <. A ,  B >. )  =  { f  e.  M  |  ( ( D `  f )  =  A  /\  ( C `  f )  =  B ) } )
 
Theoremishomc 25142 The predicate  F  e.  ( ( hom `  T
) `  <. A ,  B >. ) JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  T  e.  Cat OLD   =>    |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H ` 
 <. A ,  B >. )  <-> 
 ( F  e.  M  /\  ( D `  F )  =  A  /\  ( C `  F )  =  B ) ) )
 
Theoremishomd 25143 The predicate  F  e.  ( ( hom `  T
) `  <. A ,  B >. ) JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  H  =  ( hom `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O  /\  B  e.  O ) 
 ->  ( F  e.  ( H `  <. A ,  B >. )  <->  ( F  e.  M  /\  ( D `  F )  =  A  /\  ( C `  F )  =  B )
 ) )
 
Theoremehm 25144 The elements of a homset are morphisms. JFM CAT1 th. 21. (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   &    |-  H  =  ( hom `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O  /\  B  e.  O ) 
 ->  ( F  e.  ( H `  <. A ,  B >. )  ->  F  e.  M ) )
 
Theoremdehm 25145 Domain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  H  =  ( hom `  T )   =>    |-  (
 ( T  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H `  <. A ,  B >. )  ->  ( D `  F )  =  A ) )
 
Theoremcehm 25146 Codomain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  H  =  ( hom `  T )   =>    |-  (
 ( T  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H `  <. A ,  B >. )  ->  ( C `  F )  =  B ) )
 
Theoremmrdmcd 25147 A morphism belongs to the homset between its domain and its codomain. JFM CAT1 th. 22. (Contributed by FL, 1-Nov-2007.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Cat OLD  ->  ( F  e.  M  ->  F  e.  ( H `  <. ( D `  F ) ,  ( C `  F ) >. ) ) )
 
Theoremeqidob 25148 When the identities are equal, the objects are equal. JFM CAT1 th. 45. (Contributed by FL, 24-Apr-2007.)
 |-  O  =  dom  J   &    |-  J  =  ( id_ `  C )   =>    |-  (
 ( C  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( ( J `  A )  =  ( J `  B )  ->  A  =  B )
 )
 
Theoremhomib 25149 The homset which  ( ( id_ `  T ) `  A
) belongs to. JFM CAT1 th. 55. (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  H  =  ( hom `  T )   =>    |-  (
 ( T  e.  Cat OLD  /\  A  e.  O ) 
 ->  ( J `  A )  e.  ( H ` 
 <. A ,  A >. ) )
 
Theoremhine 25150 The homset  ( H `  <. A ,  A >. ) is not empty. JFM CAT1 th. 56. (Contributed by FL, 3-Jan-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O )  ->  ( H `  <. A ,  A >. )  =/=  (/) )
 
Theoremcmphmia 25151 Composite of the member of a homset with the identity. JFM CAT1 th. 57 (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O  /\  B  e.  O ) 
 ->  ( F  e.  ( H `  <. A ,  B >. )  ->  ( ( J `  B ) R F )  =  F ) )
 
Theoremcmphmib 25152 Composite of a member of a homset with the identity. JFM CAT1 th. 58 (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O  /\  B  e.  O ) 
 ->  ( F  e.  ( H `  <. A ,  B >. )  ->  ( F R ( J `  A ) )  =  F ) )
 
Theoremiri 25153 Composite of an identity with itself. JFM CAT1 th. 59 (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O )  ->  ( ( J `
  A ) R ( J `  A ) )  =  ( J `  A ) )
 
Theoremcmpassoh 25154  o_ is associative. Homset-based version of cmpasso 25126. (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  /\  D  e.  O ) )  ->  ( ( L  e.  ( H `
  <. A ,  B >. )  /\  M  e.  ( H `  <. B ,  C >. )  /\  N  e.  ( H `  <. C ,  D >. ) )  ->  ( N R ( M R L ) )  =  ( ( N R M ) R L ) ) )
 
Theoremhomgrf 25155 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 >. ) ) )
 
16.12.50  Monomorphisms, Epimorphisms, Isomorphisms
 
SyntaxcepiOLD 25156 Extend class notation with the class of all epimorphisms.
 class Epic
 
SyntaxcmonOLD 25157 Extend class notation with the class of all monomorphisms.
 class MonoOLD
 
SyntaxcisoOLD 25158 Extend class notation with the class of all isomorphisms.
 class  Iso OLD
 
Definitiondf-monOLD 25159* 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 25160* 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 25161* 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 25162* 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 25163* 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 25164* 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 25165 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 25166* Lemma for ismonc 25167. (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 25167* 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 25168 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 25169 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 25170 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 25171 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 25172* 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 25173* 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 25174* 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 25175 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 25176* Lemma for isepic 25177. (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 25177* 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 25178* 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 25179 Extend class notation to include a function that returns the inverses of a morphism.
 class  cinv OLD
 
Definitiondf-cinv 25180* 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 25181* 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 25182* 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 25183 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 ) ) ) )
 
16.12.51  Functors
 
SyntaxcfuncOLD 25184 Extend class notation with the class of all functors.
 class  Func OLD
 
Syntaxcifunc 25185 Extend class notation with the class of all isomorphisms.
 class  Isofunc
 
Definitiondf-funcOLD 25186* 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 25187* 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 25188* 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 25189 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 25190* The functor  F associates every object of  T to an object in  U. For the identification of objects with the identities see df-funcOLD 25186. 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 25191* 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 25192* 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 25193* 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 25194 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 25195* 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 ) ) ) } )
 
16.12.52  Subcategories
 
Syntaxcsubcat 25196 Extend class notation with a function returning all the subcategories of a given category.
 class  SubCat
 
Definitiondf-subcat 25197  (  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 25198 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 25199 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 25200 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 ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31305
  Copyright terms: Public domain < Previous  Next >