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Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisfuna 25201* 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 25202* 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 25203 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 25204* The functor  F associates every object of  T to an object in  U. For the identification of objects with the identities see df-funcOLD 25200. 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 25205* 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 25206* 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 25207* 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 25208 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 25209* 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 25210 Extend class notation with a function returning all the subcategories of a given category.
 class  SubCat
 
Definitiondf-subcat 25211  (  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 25212 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 25213 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 25214 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 25215 Lemma to simplify some subcategories related theorems . (Contributed by FL, 17-Sep-2009.)
 |-  ( U  e.  (  SubCat  `  T )  ->  T  e.  Cat
 OLD  )
 
Theoremcatsbc 25216 A category belongs to the set of its subcategories. (Contributed by FL, 17-Sep-2009.)
 |-  ( T  e.  Cat OLD  ->  T  e.  (  SubCat  `  T ) )
 
Theoremobsubc2 25217 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 25218 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 25219 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 25220 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 25221 A subcategory is a category. (Contributed by FL, 17-Sep-2009.)
 |-  ( U  e.  (  SubCat  `  T )  ->  U  e.  Cat
 OLD  )
 
Theoremmorsubc 25222 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 25223 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 25224* 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 25225 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 25226 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 25227 Extend class notation with the class of all initial objects.
 class  InitObj
 
Definitiondf-inob 25228* 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 25229* 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 25230 Extend class notation with the class of all terminal objects.
 class  TermObj
 
Definitiondf-termob 25231* 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 25232 Extend class notation with the class of all sources.
 class  Source
 
Definitiondf-source 25233* 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 25234* 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 25235* 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 25236 Extend class notation with the class of all sinks.
 class  Sink
 
Definitiondf-sink 25237* 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 25238 Extend class notation with the class of all natural sources.
 class  Natural
 
Definitiondf-natur 25239* 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 25240* 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 25241 Extend class notation with the class of all limits.
 class  LimCat
 
Definitiondf-limcat 25242* 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 25243* 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 25244 Extend class notation with the class of all object products.
 class  Prod Obj
 
Definitiondf-prodobj 25245* " 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 25246 Extend class notation with the class of all object sums.
 class  Sum Obj
 
Definitiondf-sumobj 25247* " 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 25248 Extends class notation to include function  tar.
 class  tar
 
Definitiondf-tar 25249* A function to study Tarski's classes. See valdom 25251 for its domain, vtare 25252 for its value at  (/), vtarsu 25253 for its value at a successor, vtarl 25254 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 25250* 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 25251 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 25252 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 25253* 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 25254 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 25255 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 25256 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 25257 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 25258 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 25259 Extend class notation to include the function whose value is the transitive closure of its operand.
 class  tr
 
Definitiondf-trcls 25260* 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 25261* 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 25262* 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 25263 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 25264 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 25265 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 25266 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 25267 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 25268 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 )
 
Theoremcarinttar 25269 The cardinal of the intersection of a Tarski's class with the class of the ordinal numbers. (Contributed by FL, 20-Apr-2011.)
 |-  ( T  e.  Tarski  ->  ( card `  ( On  i^i  T ) )  =  ( On  i^i  T ) )
 
Theoremcarinttar2 25270 The cardinal of a Tarski's class equals the intersection of the Tarski's class with the class of the ordinal numbers. CLASSES2 th. 10. (Contributed by FL, 20-Apr-2011.)
 |-  ( T  e.  Tarski  ->  ( card `  T )  =  ( On  i^i  T ) )
 
Theoremcardtar 25271 The cardinal of an element of a Tarski's class belongs to the Tarski's class. th. 12 CLASSES2 (Contributed by FL, 20-Apr-2011.)
 |-  (
 ( T  e.  Tarski  /\  A  e.  T ) 
 ->  ( card `  A )  e.  T )
 
Theoremcartarlim 25272 The cardinal of a Tarski's class is a limit ordinal. CLASSES2 th. 21. (Contributed by FL, 20-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
 |-  (
 ( T  e.  Tarski  /\  T  =/=  (/) )  ->  Lim  ( card `  T )
 )
 
Theoremelcarelcl 25273 An element of the cardinal of the Tarski's class  T is an element of  T. th. 14 CLASSES2. (Contributed by FL, 20-Nov-2011.)
 |-  (
 ( T  e.  Tarski  /\  A  e.  ( card `  T ) )  ->  A  e.  T )
 
Theoremfnctartar 25274 Consider functions whose domain  A is an element of a transitive Tarski's class  T and whose range is  T, then they are elements of  T. CLASSES2 th. 23. (Contributed by FL, 26-Sep-2011.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  (
 ( T  e.  Tarski  /\ 
 Tr  T  /\  A  e.  T )  ->  ( T  ^m  A )  C_  T )
 
Theoremfnctartar2 25275 Consider functions whose domain  A is an element and a part of a Tarski's class  T and whose range is  T, then they are elements of  T. CLASSES2 th. 23. (Contributed by FL, 27-Sep-2011.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  (
 ( T  e.  Tarski  /\  A  C_  T  /\  A  e.  T )  ->  ( T  ^m  A )  C_  T )
 
Theoremfnctartar3 25276 If the cardinal of  A of a part of  T is less than  T. a function from  A to  T is a part of  T. CLASSES2 th. 23. (Contributed by FL, 20-Nov-2011.)
 |-  (
 ( T  e.  Tarski  /\  A  C_  T  /\  ( card `  A )  e.  ( card `  T )
 )  ->  ( T  ^m  A )  C_  T )
 
18.12.58  Category Set
 
Syntaxccmrcase 25277 Extend class notation to include the morphisms of the category Set.
 class  Morphism SetCat
 
Definitiondf-morcatset 25278* The morphisms of the category Set. (
a is redundant and could be retrieved from  c.) Experimental. (Contributed by FL, 15-Sep-2013.)
 |-  Morphism SetCat  =  ( x  e.  Univ  |->  { <. <.
 a ,  b >. ,  c >.  |  (
 a  e.  x  /\  b  e.  x  /\  c  e.  ( b  ^m  a ) ) }
 )
 
Theoremprismorcsetlem 25279* Lemma for prismorcset 25281. (Contributed by FL, 15-Sep-2013.)
 |-  ( U  e.  Univ  ->  { <. <.
 a ,  b >. ,  c >.  |  (
 a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a ) ) }  e.  _V )
 
Theoremprismorcsetlemb 25280* Lemma for prismorcset 25281. First use of the property of a universe through grumap 8398. (Contributed by FL, 6-Nov-2013.)
 |-  ( U  e.  Univ  ->  { c  |  E. a  e.  U  E. b  e.  U  c  e.  ( b  ^m  a ) }  e.  _V )
 
Theoremprismorcset 25281 The predicate "is a morphism of the category Set". (Contributed by FL, 15-Sep-2013.)
 |-  (
 ( ( A  e.  D  /\  B  e.  E  /\  C  e.  F ) 
 /\  U  e.  Univ ) 
 ->  ( <. <. A ,  B >. ,  C >.  e.  ( Morphism SetCat `  U )  <->  ( A  e.  U  /\  B  e.  U  /\  C  e.  ( B 
 ^m  A ) ) ) )
 
Theoremmorcatset1 25282* The morphisms of the category Set. (Contributed by FL, 6-Nov-2013.)
 |-  ( U  e.  Univ  ->  ( Morphism SetCat `  U )  =  { <.
 <. a ,  b >. ,  c >.  |  (
 a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a ) ) }
 )
 
Theoremdfiunv2 25283* Define double indexed union. (Contributed by FL, 6-Nov-2013.)
 |-  U_ x  e.  A  U_ y  e.  B  C  =  {
 z  |  E. x  e.  A  E. y  e.  B  z  e.  C }
 
Theoremprismorcsetlemc 25284* Lemma for morexcmp 25334. (Contributed by FL, 6-Nov-2013.)
 |-  F  =  { <. <. a ,  b >. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a ) ) }   =>    |-  ( U  e.  Univ  ->  F  C_  ( ( U  X.  U )  X.  U_ a  e.  U  U_ b  e.  U  ( b  ^m  a ) ) )
 
Theoremprismorcset2 25285 The predicate "is a morphism of the category Set". (Contributed by FL, 15-Sep-2013.)
 |-  A  =  ( ( 1st  o.  1st ) `  M )   &    |-  B  =  ( ( 2nd  o.  1st ) `  M )   &    |-  C  =  ( 2nd `  M )   =>    |-  (
 ( U  e.  Univ  /\  M  e.  ( Morphism SetCat `  U ) )  ->  ( A  e.  U  /\  B  e.  U  /\  C  e.  ( B  ^m  A ) ) )
 
Syntaxcdomcase 25286 Extend class notation to include the domain of a morphism in the category Set.
 class  dom SetCat
 
Definitiondf-domcatset 25287* The domain of a morphism in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)
 |-  dom SetCat  =  ( x  e.  Univ  |->  ( a  e.  ( Morphism SetCat `  x )  |->  ( ( 1st  o.  1st ) `  a ) ) )
 
Syntaxcgraphcase 25288 Extend class notation to include the graph of a morphism in the category Set.
 class  graph SetCat
 
Definitiondf-graphcatset 25289* The underlying function of a morphism in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)
 |-  graph SetCat  =  ( x  e.  Univ  |->  ( a  e.  ( Morphism SetCat `  x )  |->  ( 2nd `  a
 ) ) )
 
Theoremisgraphmrph 25290 The graph of a morhism in the category Set. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U ) )  ->  ( ( graph SetCat `  U ) `  A )  =  ( 2nd `  A ) )
 
Theoremisgraphmrph2 25291 The graph of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
 |- .graph  =  ( graph SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   =>    |-  (
 ( U  e.  Univ  /\  A  e. .Morphism  )  ->  (.graph  `  A )  =  ( 2nd `  A ) )
 
Theoremdomcatfun 25292 The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
 |-  ( U  e.  Univ  ->  ( dom
 SetCat `  U ) : ( Morphism SetCat `  U ) --> U )
 
Theoremdomdomcatfun 25293 The domain of the function  dom SetCat in the category Set. (Contributed by FL, 6-Nov-2013.)
 |-  ( U  e.  Univ  ->  dom  ( dom
 SetCat `  U )  =  ( Morphism SetCat `  U )
 )
 
Theoremdomdomcatfun1 25294 The domain of the function  dom SetCat in the category Set. (Contributed by FL, 6-Nov-2013.)
 |- .dom  =  ( dom SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   =>    |-  ( U  e.  Univ  ->  dom .dom  = .Morphism  )
 
Theoremdomcatsetval 25295 The domain of a morphism in the category Set is a member of the underlying universe. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U ) )  ->  ( ( dom SetCat `  U ) `  A )  e.  U )
 
Theoremdomcatsetval2 25296 The domain of a morphism in the category Set is a member of the underlying universe. (Contributed by FL, 6-Nov-2013.)
 |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   =>    |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  (.dom  `  F )  e.  U )
 
Theoremdomcatval 25297 The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U ) )  ->  ( ( dom SetCat `  U ) `  A )  =  ( ( 1st  o.  1st ) `  A ) )
 
Theoremdomcatval2 25298 The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
 |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   =>    |-  ( ( U  e.  Univ  /\  A  e. .Morphism  )  ->  (.dom  `  A )  =  ( ( 1st  o.  1st ) `  A ) )
 
Syntaxccodcase 25299 Extend class notation to include the codomain of a morphism in the category Set.
 class  cod SetCat
 
Definitiondf-codcatset 25300* The codomain of a morphism in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)
 |-  cod SetCat  =  ( x  e.  Univ  |->  ( a  e.  ( Morphism SetCat `  x )  |->  ( ( 2nd  o.  1st ) `  a ) ) )
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