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Theorem List for Metamath Proof Explorer - 26001-26100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtareltsuc 26001 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 26002 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 26003 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 26004 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 26005 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 26006 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 26007 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 26008 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 26009 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 26010 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 26011 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 26012 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 )
 
18.13.58  Category Set
 
Syntaxccmrcase 26013 Extend class notation to include the morphisms of the category Set.
 class  Morphism SetCat
 
Definitiondf-morcatset 26014* 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 26015* Lemma for prismorcset 26017. (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 26016* Lemma for prismorcset 26017. First use of the property of a universe through grumap 8446. (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 26017 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 26018* 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 26019* 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 26020* Lemma for morexcmp 26070. (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 26021 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 26022 Extend class notation to include the domain of a morphism in the category Set.
 class  dom SetCat
 
Definitiondf-domcatset 26023* 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 26024 Extend class notation to include the graph of a morphism in the category Set.
 class  graph SetCat
 
Definitiondf-graphcatset 26025* 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 26026 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 26027 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 26028 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 26029 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 26030 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 26031 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 26032 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 26033 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 26034 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 26035 Extend class notation to include the codomain of a morphism in the category Set.
 class  cod SetCat
 
Definitiondf-codcatset 26036* 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 ) ) )
 
Theoremcodcatfun 26037 The codomain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
 |-  ( U  e.  Univ  ->  ( cod
 SetCat `  U ) : ( Morphism SetCat `  U ) --> U )
 
Theoremcodcatsetval 26038 The codomain 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 ) )  ->  ( ( cod SetCat `  U ) `  A )  e.  U )
 
Theoremcodcatval 26039 The codomain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U ) )  ->  ( ( cod SetCat `  U ) `  A )  =  ( ( 2nd  o.  1st ) `  A ) )
 
Theoremcodcatval2 26040 The codomain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
 |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .cod  =  ( cod SetCat `  U )   =>    |-  ( ( U  e.  Univ  /\  A  e. .Morphism  )  ->  (.cod  `  A )  =  ( ( 2nd  o.  1st ) `  A ) )
 
Theoremprismorcset3 26041 The predicate "is a morphism of the category Set". (Contributed by FL, 6-Nov-2013.)
 |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   &    |- .graph  =  ( graph SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   =>    |-  (
 ( U  e.  Univ  /\  M  e. .Morphism  )  ->  (.graph  `  M )  e.  ( (.cod  `  M )  ^m  (.dom  `  M ) ) )
 
Syntaxcidcase 26042 Extend class notation to include the identity morphism of an object in the category Set.
 class  Id SetCat
 
Definitiondf-idcatset 26043* The identity morphims in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)
 |-  Id SetCat  =  ( x  e.  Univ  |->  ( a  e.  x  |->  <. <. a ,  a >. ,  (  _I  |`  a )
 >. ) )
 
Theoremidcatfun 26044 The identity morphims in the category Set. (Contributed by FL, 6-Nov-2013.)
 |-  ( U  e.  Univ  ->  ( Id
 SetCat `  U ) : U --> ( Morphism SetCat `  U ) )
 
Theoremobcatset 26045 The objects of the category Set is the Universe. (Contributed by FL, 6-Nov-2013.)
 |- .Object  =  dom  ( Id SetCat `  U )   =>    |-  ( U  e.  Univ  -> .Object  =  U )
 
Theoremidcatval 26046 An identity morphism is a morphism. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  U ) 
 ->  ( ( Id SetCat `  U ) `  A )  e.  ( Morphism SetCat `  U )
 )
 
Theoremidcatval2 26047 An identity morphism is a morphism. (Contributed by FL, 6-Nov-2013.)
 |- .id  =  ( Id SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   =>    |-  (
 ( U  e.  Univ  /\  A  e.  U ) 
 ->  (.id  `  A )  e. .Morphism  )
 
Theoremdomidcat 26048 The underlying universe of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |-  ( U  e.  Univ  ->  dom  ( Id
 SetCat `  U )  =  U )
 
Theoremidmor 26049 An identity morphism. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  U ) 
 ->  ( ( Id SetCat `  U ) `  A )  = 
 <. <. A ,  A >. ,  (  _I  |`  A )
 >. )
 
Theoremidmorimor 26050 An identity morphism is a morphism. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  U ) 
 ->  <. <. A ,  A >. ,  (  _I  |`  A )
 >.  e.  ( Morphism SetCat `  U ) )
 
Theoremdomidmor 26051 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  U ) 
 ->  ( ( dom SetCat `  U ) `  ( ( Id SetCat `
  U ) `  A ) )  =  A )
 
Theoremdomidmor2 26052 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |- .dom  =  ( dom SetCat `  U )   &    |- .id  =  ( Id SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (.dom  `  (.id  `  A ) )  =  A )
 
Theoremcodidmor 26053 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  U ) 
 ->  ( ( cod SetCat `  U ) `  ( ( Id SetCat `
  U ) `  A ) )  =  A )
 
Theoremcodidmor2 26054 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |- .cod  =  ( cod SetCat `  U )   &    |- .id  =  ( Id SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (.cod  `  (.id  `  A ) )  =  A )
 
Theoremgrphidmor 26055 Graph of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  U ) 
 ->  ( ( graph SetCat `  U ) `  ( ( Id SetCat `
  U ) `  A ) )  =  (  _I  |`  A ) )
 
Theoremgrphidmor2 26056 Graph of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |- .graph  =  2nd   &    |- .id  =  ( Id SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (.graph  `  (.id  `  A ) )  =  (  _I  |`  A ) )
 
Theoremgrphidmor3 26057 Graph of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |- .graph  =  ( graph SetCat `  U )   &    |- .id  =  ( Id SetCat `  U )   =>    |-  ( ( U  e.  Univ  /\  A  e.  U ) 
 ->  (.graph  `  (.id  `  A ) )  =  (  _I  |`  A ) )
 
Syntaxcrocase 26058 Extend class notation to include the morphisms composition in the category Set.
 class  ro SetCat
 
Definitiondf-rocatset 26059* Composition of two morphisms in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)
 |-  ro SetCat  =  ( x  e.  Univ  |->  {
 <. <. a ,  b >. ,  c >.  |  ( ( a  e.  ( Morphism SetCat `  x )  /\  b  e.  ( Morphism SetCat `  x )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
 o.  1st ) `  b
 ) )  /\  c  =  <. <. ( ( 1st 
 o.  1st ) `  b
 ) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  (
 ( 2nd `  a )  o.  ( 2nd `  b
 ) ) >. ) }
 )
 
Theoremisrocatset 26060* Definition of the composition of two morphisms in the category Set . (Contributed by FL, 6-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  ( U  e.  Univ  ->  ( ro
 SetCat `  U )  =  { <. <. a ,  b >. ,  c >.  |  ( ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
 o.  1st ) `  b
 ) )  /\  c  =  <. <. ( ( 1st 
 o.  1st ) `  b
 ) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  (
 ( 2nd `  a )  o.  ( 2nd `  b
 ) ) >. ) }
 )
 
Theoremcmp2morp 26061 Composite of two morphisms. (Contributed by FL, 6-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  O  =  ( ro SetCat `  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
 )  /\  ( ( dom
 SetCat `  U ) `  A )  =  (
 ( cod SetCat `  U ) `  B ) )  ->  ( A O B )  =  <. <. ( ( 1st 
 o.  1st ) `  B ) ,  ( ( 2nd  o.  1st ) `  A ) >. ,  (
 ( 2nd `  A )  o.  ( 2nd `  B ) ) >. )
 
Theoremrocatval 26062 The composite of two morphisms in the category Set is a morphism. (Contributed by FL, 6-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
 )  /\  ( ( dom
 SetCat `  U ) `  A )  =  (
 ( cod SetCat `  U ) `  B ) )  ->  ( A O B )  e.  ( Morphism SetCat `  U ) )
 
Theoremrocatval2 26063 The composite of two morphisms in the category Set is a morphism. (Contributed by FL, 7-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e. .Morphism  )
 
Theoremcmp2morpcats 26064 Composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <.
 (.dom  `  B ) ,  (.cod  `  A ) >. ,  (
 ( 2nd `  A )  o.  ( 2nd `  B ) ) >. )
 
Theoremcmp2morpcatt 26065 Composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   &    |- .graph  =  ( graph SetCat `  U )   =>    |-  (
 ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <. (.dom  `  B ) ,  (.cod  `  A ) >. ,  (
 (.graph  `  A )  o.  (.graph  `  B ) ) >. )
 
Theoremcmp2morpgrp 26066 Graph of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   &    |- .graph  =  ( graph SetCat `  U )   =>    |-  (
 ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.graph  `  ( A O B ) )  =  ( (.graph  `  A )  o.  (.graph  `  B ) ) )
 
Theoremcmp2morpdom 26067 Domain of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.dom  `  ( A O B ) )  =  (.dom  `  B ) )
 
Theoremcmp2morpcod 26068 Codomain of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.cod  `  ( A O B ) )  =  (.cod  `  A ) )
 
Theoremcmpmorass 26069 Associativity of composition in category Set. (Contributed by FL, 7-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B ) 
 /\  (.dom  `  B )  =  (.cod  `  A ) ) ) 
 ->  ( C O ( B O A ) )  =  ( ( C O B ) O A ) )
 
Theoremmorexcmp 26070 A morphism expressed thanks to its components. (Contributed by FL, 8-Nov-2013.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   &    |- .graph  =  2nd   =>    |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. (.dom  `  F ) ,  (.cod  `  F ) >. ,  (.graph  `  F ) >. )
 
Theoremmorexcmp2 26071 A morphism expressed thanks to its components. (Contributed by FL, 8-Nov-2013.)
 |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   &    |- .graph  =  ( graph SetCat `  U )   =>    |-  (
 ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. (.dom  `  F ) ,  (.cod  `  F ) >. ,  (.graph  `  F ) >. )
 
Theoremcmpidmor2 26072 Composition with an identity. (Contributed by FL, 8-Nov-2013.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .cod  =  ( cod SetCat `  U )   &    |- .id  =  ( Id SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( (.id  `  (.cod  `  F ) ) O F )  =  F )
 
Theoremcmpidmor3 26073 Composition with an identity. (Contributed by FL, 8-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .id  =  ( Id SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( F O (.id  `  (.dom  `  F ) ) )  =  F )
 
Theoremcmpmorfun 26074 Composition of morphisms is a function. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  ( U  e.  Univ  ->  Fun  ( ro
 SetCat `  U ) )
 
Theoremcmppar2 26075* Morphisms composition is defined every time the codomain of the second operand matches the domain of the first one. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  ( U  e.  Univ  ->  dom  ( ro
 SetCat `  U )  =  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
 ( 1st  o.  1st ) `  a )  =  ( ( 2nd  o.  1st ) `  b ) ) } )
 
Theoremcmppar 26076 Composition of morphisms is a partial operation in the set of morphisms. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  ( U  e.  Univ  ->  dom  ( ro
 SetCat `  U )  C_  ( ( Morphism SetCat `  U )  X.  ( Morphism SetCat `  U ) ) )
 
Theoremcmppar3 26077 Morphisms composition is defined every time the codomain of the second operand matches the domain of the first one. (Contributed by FL, 8-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
 )  ->  ( <. A ,  B >.  e.  dom  ( ro SetCat `  U )  <->  ( ( dom SetCat `  U ) `  A )  =  ( ( cod SetCat `  U ) `  B ) ) )
 
Theoremcmpmor 26078 The composite of two morphisms is a morphism. (Contributed by FL, 8-Nov-2013.)
 |-  ( U  e.  Univ  ->  ran  ( ro
 SetCat `  U )  C_  ( Morphism SetCat `  U )
 )
 
Syntaxccaset 26079 Extend class notation to include the category Set.
 class  SetCat OLD
 
Definitiondf-catset 26080 Definition of the category Set. (We should say "the categories Set" since there is such a category per universe but for our purpose they are equivalent obviously.) Experimental. (Contributed by FL, 8-Nov-2013.)
 |-  SetCat OLD  =  ( x  e.  Univ  |->  <. <. ( dom SetCat `
  x ) ,  ( cod SetCat `  x ) >. ,  <. ( Id SetCat `
  x ) ,  ( ro SetCat `  x ) >. >. )
 
Theoremiscatset 26081 The category Set. (Contributed by FL, 8-Nov-2013.)
 |-  ( U  e.  Univ  ->  ( SetCat OLD `  U )  =  <. <. ( dom SetCat `  U ) ,  ( cod SetCat `  U ) >. ,  <. ( Id SetCat `  U ) ,  ( ro SetCat `  U ) >. >. )
 
Theoremsetiscat 26082 The category set is a category. (Contributed by FL, 6-Nov-2013.)
 |-  ( U  e.  Univ  ->  ( SetCat OLD `  U )  e.  Cat OLD  )
 
18.13.59  Grammars, Logics, Machines and Automata
 
Syntaxckln 26083 Extend class notation with the Kleene star.
 class  Kleene
 
18.13.60  Words
 
Syntaxcwrd 26084 Extend class notation with the class of words of a given size.
 class  Words
 
Definitiondf-words 26085* The words of size  y over an alphabet  x are the finite sequences over  x of size  y. Their domains are fiercely set to  ( 1 ... y ) so that I can concatenate them easily. The case  y  =  0 is a bit tricky and corresponds to the unique empty word (often denoted by an epsilon or by  1 in textbooks.) Experimental. (Contributed by FL, 14-Jan-2014.)
 |-  Words  =  ( x  e.  _V ,  y  e.  NN0  |->  ( x 
 ^m  ( 1 ... y ) ) )
 
Theoremisword 26086 The words over a set  A. (Contributed by FL, 14-Jan-2014.)
 |-  (
 ( A  e.  B  /\  N  e.  NN0 )  ->  ( A  Words  N )  =  ( A  ^m  ( 1 ... N ) ) )
 
Syntaxcdwords 26087 Syntax for the dWords operator.
 class dWords
 
Definitiondf-dwords 26088* Words of size S over an alphabet  A with all the elements different. (For my private use only. Don't use.) (Contributed by FL, 26-May-2016.)
 |- dWords  =  ( a  e.  _V ,  s  e.  NN0  |->  { w  e.  ( a  Words  s )  |  A. x  e.  ( 1 ... s
 ) A. y  e.  (
 1 ... s ) ( x  =/=  y  ->  ( w `  x )  =/=  ( w `  y ) ) }
 )
 
Theoremisnword 26089* The words over a set  A. (For my private use only. Don't use.) (Contributed by FL, 26-May-2014.)
 |-  (
 ( A  e.  B  /\  S  e.  NN0 )  ->  ( W  e.  ( AdWords S )  <->  ( W  e.  ( A  ^m  ( 1
 ... S ) ) 
 /\  A. x  e.  (
 1 ... S ) A. y  e.  ( 1 ... S ) ( x  =/=  y  ->  ( W `  x )  =/=  ( W `  y
 ) ) ) ) )
 
Definitiondf-kle 26090* The Kleene star of an alphabet  x is the set of all the finite sequences of elements of this alphabet. Experimental. (Contributed by FL, 14-Jan-2014.)
 |-  Kleene  =  ( x  e.  _V  |->  U_ u  e.  NN0  ( x 
 ^m  ( 1 ... u ) ) )
 
TheoremisKleene 26091* The predicate is the Kleene star of a set  A. An element of  ( Kleene `  A
) is called a word. (Contributed by FL, 14-Jan-2014.)
 |-  ( A  e.  B  ->  (
 Kleene `  A )  = 
 U_ u  e.  NN0  ( A  ^m  ( 1
 ... u ) ) )
 
Theorem1iskle 26092 Symbols and variables belong to the Kleene star of  NN. (Contributed by FL, 2-Feb-2014.)
 |-  ( A  e.  NN  ->  {
 <. 1 ,  A >. }  e.  ( Kleene `  NN ) )
 
Theoremselsubf 26093 A way of selecting a subset of functions so that their values belong to  B. (Contributed by FL, 14-Jan-2014.)
 |-  A  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  ^m  C )  i^i  ~P ( C  X.  B ) )  =  ( ( A  i^i  B )  ^m  C )
 
Theoremselsubf3 26094 A way of selecting a subset of functions so that their values belong to  B. (Contributed by FL, 14-Jan-2014.)
 |-  A  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  ^m  C )  i^i  ~P ( _V 
 X.  B ) )  =  ( ( A  i^i  B )  ^m  C )
 
Theoremselsubf3g 26095 A way of selecting a subset of functions so that their values belong to  B. (Contributed by FL, 14-Jan-2014.)
 |-  (
 ( A  e.  D  /\  C  e.  E ) 
 ->  ( ( A  ^m  C )  i^i  ~P ( _V  X.  B ) )  =  ( ( A  i^i  B )  ^m  C ) )
 
Syntaxclincl 26096 Extend class notation with the class of inductive closures.
 class  IndCls
 
Definitiondf-indcls 26097* Definition of an inductive closure. Top down definition. Gallier p. 19 (Contributed by FL, 14-Jan-2014.)
 |-  IndCls  =  ( x  e.  _V ,  y  e.  _V  |->  |^| { a  |  ( x  C_  a  /\  A. f  e.  y  A. j  e.  ( dom  f  i^i  ~P ( _V  X.  a ) ) ( f `  j
 )  e.  a ) } )
 
Theoremlemindclsbu 26098* Lemma for indcls2 26099. (Contributed by FL, 14-Jan-2014.)
 |-  (
 ( U  e.  A  /\  A. f  e.  F  E. n  e.  NN  f  e.  ( U  ^m  ( U  ^m  (
 1 ... n ) ) ) )  ->  F  e.  _V )
 
Theoremindcls2 26099* The inductive closure of  X under  F. (Contributed by FL, 14-Jan-2014.)
 |-  (
 ( U  e.  A  /\  X  C_  U  /\  A. f  e.  F  E. n  e.  NN  f  e.  ( U  ^m  ( U  ^m  ( 1 ... n ) ) ) )  ->  ( X  IndCls  F )  =  |^| { a  |  ( X 
 C_  a  /\  A. f  e.  F  A. j  e.  ( dom  f  i^i 
 ~P ( _V  X.  a ) ) ( f `  j )  e.  a ) }
 )
 
Theoremxindcls 26100* X is a part of the inductive closure of  X under  F. (Contributed by FL, 15-Jan-2014.)
 |-  (
 ( U  e.  A  /\  X  C_  U  /\  A. f  e.  F  E. n  e.  NN  f  e.  ( U  ^m  ( U  ^m  ( 1 ... n ) ) ) )  ->  X  C_  ( X  IndCls  F ) )
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