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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-ded 24901* Definition of a deductive system. Lambeck and Scott. Introduction to higher order categorical logic. p. 47. 1986. Unformally we can say a deductive system is a directed multi graph where for each object a specific morphism called identity of the object exists and where for some pairs of morphisms the composite exists. Deductive system are named so because morphisms may be interpreted as logical deductions, objects as sets of formulas and compositions as inferences. (Contributed by FL, 24-Oct-2007.)
 |-  Ded  =  { x  |  E. d E. c E. j E. r ( x  = 
 <. <. d ,  c >. ,  <. j ,  r >.
 >.  /\  ( ( <. <.
 d ,  c >. , 
 <. j ,  r >. >.  e.  Alg  /\  A. a  e. 
 dom  j ( ( d `  ( j `
  a ) )  =  a  /\  (
 c `  ( j `  a ) )  =  a )  /\  A. f  e.  dom  d A. g  e.  dom  d (
 <. g ,  f >.  e. 
 dom  r  <->  ( d `  g )  =  (
 c `  f )
 ) )  /\  ( A. f  e.  dom  d A. g  e.  dom  d ( ( d `
  g )  =  ( c `  f
 )  ->  ( d `  ( g r f ) )  =  ( d `  f ) )  /\  A. f  e.  dom  d A. g  e.  dom  d ( ( d `  g )  =  ( c `  f )  ->  ( c `
  ( g r f ) )  =  ( c `  g
 ) ) ) ) ) }
 
Theoremisded 24902* The predicate "is a deductive system". (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.  Ded  <->  ( ( <. <. D ,  C >. , 
 <. J ,  R >. >.  e.  Alg  /\  A. a  e.  O  ( ( D `
  ( J `  a ) )  =  a  /\  ( C `
  ( J `  a ) )  =  a )  /\  A. f  e.  M  A. g  e.  M  ( <. g ,  f >.  e.  dom  R  <-> 
 ( D `  g
 )  =  ( C `
  f ) ) )  /\  ( A. f  e.  M  A. g  e.  M  ( ( D `
  g )  =  ( C `  f
 )  ->  ( D `  ( g R f ) )  =  ( D `  f ) )  /\  A. f  e.  M  A. g  e.  M  ( ( D `
  g )  =  ( C `  f
 )  ->  ( C `  ( g R f ) )  =  ( C `  g ) ) ) ) ) )
 
Theoremdedi 24903* Properties of a deductive system. (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.  Ded  ->  ( ( <. <. D ,  C >. ,  <. J ,  R >. >.  e.  Alg  /\  A. a  e.  O  ( ( D `  ( J `  a ) )  =  a  /\  ( C `  ( J `  a ) )  =  a )  /\  A. f  e.  M  A. g  e.  M  ( <. g ,  f >.  e.  dom  R  <-> 
 ( D `  g
 )  =  ( C `
  f ) ) )  /\  ( A. f  e.  M  A. g  e.  M  ( ( D `
  g )  =  ( C `  f
 )  ->  ( D `  ( g R f ) )  =  ( D `  f ) )  /\  A. f  e.  M  A. g  e.  M  ( ( D `
  g )  =  ( C `  f
 )  ->  ( C `  ( g R f ) )  =  ( C `  g ) ) ) ) )
 
Theorem1ded 24904 Category  1 is a deductive system. We can think of the morphism of Category  1 as corresponding to  ph |-  ph. (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.  Ded
 
Theoremstrded 24905 Structure of a deductive system. (Contributed by FL, 28-Oct-2007.)
 |-  Ded  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
 
Theoremrelded 24906 A deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  Ded
 
Theoremreldded 24907 The domain of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  dom 
 Ded
 
Theoremrelrded 24908 The range of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  ran 
 Ded
 
Theoremdedalg 24909 A deductive system is an "algebra". (Contributed by FL, 28-Oct-2007.)
 |-  ( T  e.  Ded  ->  T  e.  Alg  )
 
Theoremidosd 24910 The identity is a morphism which has the same object as its domain and its codomain. (Contributed by FL, 28-Oct-2007.)
 |-  O  =  dom  J   &    |-  D  =  (
 dom_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( ( T  e.  Ded  /\  A  e.  O ) 
 ->  ( ( D `  ( J `  A ) )  =  A  /\  ( C `  ( J `
  A ) )  =  A ) )
 
Theoremcmppfd 24911  ( G
( o_ `  T
) F ) is only defined when the domain of  G is the codomain of  F. (Contributed by FL, 29-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Ded  /\  F  e.  M  /\  G  e.  M )  ->  ( <. G ,  F >.  e.  dom  R  <->  ( D `  G )  =  ( C `  F ) ) )
 
Theoremdomcmpd 24912 When  ( G
( o_ `  T
) F ) is defined its domain is the domain of  F. (Contributed by FL, 29-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Ded  /\  F  e.  M  /\  G  e.  M )  ->  ( ( D `  G )  =  ( C `  F )  ->  ( D `  ( G R F ) )  =  ( D `  F ) ) )
 
Theoremcodcmpd 24913 When  ( G
( o_ `  T
) F ) is defined its codomain is the codomain of  G. (Contributed by FL, 29-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Ded  /\  F  e.  M  /\  G  e.  M )  ->  ( ( D `  G )  =  ( C `  F )  ->  ( C `  ( G R F ) )  =  ( C `  G ) ) )
 
Theoremrdmob 24914 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 24915 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 24916 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 24917 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 ) )
 
16.11.48  Categories
 
SyntaxccatOLD 24918 Extend class notation with the class of categories.
 class  Cat OLD
 
Definitiondf-catOLD 24919* 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 24920* 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 24921* 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 24922 Lemma for 0ded 24923. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Alg
 
Theorem0ded 24923 A deductive system with no object and no morphism. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Ded
 
Theorem0catOLD 24924 Category  0 has no object and no morphism. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Cat OLD
 
Theorem1cat 24925 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 24926 Structure of a category. (Contributed by FL, 26-Oct-2007.)
 |-  Cat OLD  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
 
Theoremrelcat 24927 A category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  Cat
 OLD
 
Theoremreldcat 24928 The domain of a category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  dom 
 Cat OLD
 
Theoremrelrcat 24929 The range of a category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  ran 
 Cat OLD
 
Theoremcatded 24930 A category is a deductive system. (Contributed by FL, 26-Oct-2007.)
 |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
 
Theoremdomc 24931 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 24932 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 24933 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 24934 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 24935 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 24936 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 24937 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 24938 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 24939 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 24940 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 24941 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 24942 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 24943 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 24944 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 24945 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 24946 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 24947 Composition is a function. (Contributed by FL, 5-Oct-2009.)
 |-  R  =  ( o_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  Fun  R )
 
Theoremdualalg 24948 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 24949 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 24950 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  )
 
16.11.49  Homsets
 
SyntaxchomOLD 24951 Extend class notation with the function returning all the morphisms between two objects.
 class  hom
 
Definitiondf-homOLD 24952*  ( 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 24953* 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 24954* 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 24955 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 24956 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 24957 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 24958 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 24959 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 24960 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 24961 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 24962 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 24963 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 24964 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 24965 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 24966 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 24967  o_ is associative. Homset-based version of cmpasso 24939. (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 24968 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 >. ) ) )
 
16.11.50  Monomorphisms, Epimorphisms, Isomorphisms
 
SyntaxcepiOLD 24969 Extend class notation with the class of all epimorphisms.
 class Epic
 
SyntaxcmonOLD 24970 Extend class notation with the class of all monomorphisms.
 class MonoOLD
 
SyntaxcisoOLD 24971 Extend class notation with the class of all isomorphisms.
 class  Iso OLD
 
Definitiondf-monOLD 24972* 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 24973* 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 24974* 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 24975* 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 24976* 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 24977* 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 24978 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 24979* Lemma for ismonc 24980. (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 24980* 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 24981 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 24982 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 24983 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 24984 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 24985* 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 24986* 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 24987* 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 24988 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 24989* Lemma for isepic 24990. (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 24990* 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 24991* 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 24992 Extend class notation to include a function that returns the inverses of a morphism.
 class  cinv OLD
 
Definitiondf-cinv 24993* 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 24994* 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 24995* 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 24996 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 ) ) ) )
 
16.11.51  Functors
 
SyntaxcfuncOLD 24997 Extend class notation with the class of all functors.
 class  Func OLD
 
Syntaxcifunc 24998 Extend class notation with the class of all isomorphisms.
 class  Isofunc
 
Definitiondf-funcOLD 24999* 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 25000* 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 ) ) ) ) } )
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