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Theorem List for Metamath Proof Explorer - 13901-14000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
8.2.1  Identity and composition for arrows
 
Syntaxcida 13901 Extend class notation to include identity for arrows.
 class Ida
 
Syntaxccoa 13902 Extend class notation to include composition for arrows.
 class compa
 
Definitiondf-ida 13903* Definition of the identity arrow, which is just the identity morphism tagged with its domain and codomain. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
 |- Ida  =  ( c  e.  Cat  |->  ( x  e.  ( Base `  c )  |->  <. x ,  x ,  ( ( Id `  c ) `  x ) >. ) )
 
Definitiondf-coa 13904* Definition of the composition of arrows. Since arrows are tagged with domain and codomain, this does not need to be a 5-ary operation like the regular composition in a category comp. Instead, it is a partial binary operation on arrows, which is defined when the domain of the first arrow matches the codomain of the second. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- compa  =  ( c  e.  Cat  |->  ( g  e.  (Nat `  c ) ,  f  e.  { h  e.  (Nat `  c )  |  (coda `  h )  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g
 ) ,  ( ( 2nd `  g )
 ( <. (domA `  f ) ,  (domA `  g )
 >. (comp `  c )
 (coda `  g ) ) ( 2nd `  f )
 ) >. ) )
 
Theoremidafval 13905* Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  I  =  (Ida `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  .1.  =  ( Id `  C )   =>    |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. ) )
 
Theoremidaval 13906 Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  I  =  (Ida `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( I `  X )  = 
 <. X ,  X ,  (  .1.  `  X ) >. )
 
Theoremida2 13907 Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  I  =  (Ida `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( 2nd `  ( I `  X ) )  =  (  .1.  `  X ) )
 
Theoremidahom 13908 Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  I  =  (Ida `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  H  =  (Homa `  C )   =>    |-  ( ph  ->  ( I `  X )  e.  ( X H X ) )
 
Theoremidadm 13909 Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  I  =  (Ida `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (domA `  ( I `
  X ) )  =  X )
 
Theoremidacd 13910 Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  I  =  (Ida `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (coda `  ( I `  X ) )  =  X )
 
Theoremidaf 13911 The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  I  =  (Ida `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  A  =  (Nat `  C )   =>    |-  ( ph  ->  I : B --> A )
 
Theoremcoafval 13912* The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  A  =  (Nat `  C )   &    |-  .xb  =  (comp `  C )   =>    |- 
 .x.  =  ( g  e.  A ,  f  e. 
 { h  e.  A  |  (coda `  h )  =  (domA `  g
 ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  (
 ( 2nd `  g )
 ( <. (domA `  f ) ,  (domA `  g )
 >.  .xb  (coda `  g ) ) ( 2nd `  f )
 ) >. )
 
Theoremeldmcoa 13913 A pair  <. G ,  F >. is in the domain of the arrow composition, if the domain of  G equals the codomain of  F. (In this case we say  G and  F are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  A  =  (Nat `  C )   =>    |-  ( G dom  .x.  F  <->  ( F  e.  A  /\  G  e.  A  /\  (coda `  F )  =  (domA `  G ) ) )
 
Theoremdmcoass 13914 The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  A  =  (Nat `  C )   =>    |- 
 dom  .x.  C_  ( A  X.  A )
 
Theoremhomdmcoa 13915 If  F : X --> Y and  G : Y --> Z, then  G and  F are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  H  =  (Homa `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   =>    |-  ( ph  ->  G dom  .x. 
 F )
 
Theoremcoaval 13916 Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  H  =  (Homa `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   &    |-  .xb 
 =  (comp `  C )   =>    |-  ( ph  ->  ( G  .x.  F )  = 
 <. X ,  Z ,  ( ( 2nd `  G ) ( <. X ,  Y >.  .xb  Z ) ( 2nd `  F )
 ) >. )
 
Theoremcoa2 13917 The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  H  =  (Homa `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   &    |-  .xb 
 =  (comp `  C )   =>    |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( ( 2nd `  G ) ( <. X ,  Y >.  .xb  Z ) ( 2nd `  F )
 ) )
 
Theoremcoahom 13918 The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  H  =  (Homa `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   =>    |-  ( ph  ->  ( G  .x.  F )  e.  ( X H Z ) )
 
Theoremcoapm 13919 Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  A  =  (Nat `  C )   =>    |- 
 .x.  e.  ( A  ^pm  ( A  X.  A ) )
 
Theoremarwlid 13920 Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |- 
 .x.  =  (compa `  C )   &    |-  .1.  =  (Ida `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( (  .1.  `  Y )  .x.  F )  =  F )
 
Theoremarwrid 13921 Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |- 
 .x.  =  (compa `  C )   &    |-  .1.  =  (Ida `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( F  .x.  (  .1.  `  X ) )  =  F )
 
Theoremarwass 13922 Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |- 
 .x.  =  (compa `  C )   &    |-  .1.  =  (Ida `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   &    |-  ( ph  ->  K  e.  ( Z H W ) )   =>    |-  ( ph  ->  (
 ( K  .x.  G )  .x.  F )  =  ( K  .x.  ( G  .x.  F ) ) )
 
8.3  Examples of categories
 
8.3.1  The category of sets
 
Syntaxcsetc 13923 Extend class notation to include the category Set.
 class  SetCat
 
Definitiondf-setc 13924* Definition of the category Set, relativized to a subset  u. This is the category of all sets in 
u and functions between these sets. Generally, we will take  u to be a weak universe or Grothendieck's universe, because these sets have closure properties as good as the real thing. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 3-Jan-2017.)
 |-  SetCat  =  ( u  e. 
 _V  |->  { <. ( Base `  ndx ) ,  u >. , 
 <. (  Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( y  ^m  x ) ) >. , 
 <. (comp `  ndx ) ,  ( v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( z 
 ^m  ( 2nd `  v
 ) ) ,  f  e.  ( ( 2nd `  v
 )  ^m  ( 1st `  v ) )  |->  ( g  o.  f ) ) ) >. } )
 
Theoremsetcval 13925* Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( y 
 ^m  x ) ) )   &    |-  ( ph  ->  .x. 
 =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  (
 ( 2nd `  v )  ^m  ( 1st `  v
 ) )  |->  ( g  o.  f ) ) ) )   =>    |-  ( ph  ->  C  =  { <. ( Base `  ndx ) ,  U >. , 
 <. (  Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .x.  >. } )
 
Theoremsetcbas 13926 Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  U  =  ( Base `  C ) )
 
Theoremsetchomfval 13927* Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  (  Hom  `  C )   =>    |-  ( ph  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( y  ^m  x ) ) )
 
Theoremsetchom 13928 Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( X H Y )  =  ( Y  ^m  X ) )
 
Theoremelsetchom 13929 A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( F  e.  ( X H Y )  <->  F : X --> Y ) )
 
Theoremsetccofval 13930* Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  .x.  =  (comp `  C )   =>    |-  ( ph  ->  .x. 
 =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  (
 ( 2nd `  v )  ^m  ( 1st `  v
 ) )  |->  ( g  o.  f ) ) ) )
 
Theoremsetcco 13931 Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  F : X --> Y )   &    |-  ( ph  ->  G : Y --> Z )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  =  ( G  o.  F ) )
 
Theoremsetccatid 13932* Lemma for setccat 13933. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   =>    |-  ( U  e.  V  ->  ( C  e.  Cat  /\  ( Id `  C )  =  ( x  e.  U  |->  (  _I  |`  x ) ) ) )
 
Theoremsetccat 13933 The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   =>    |-  ( U  e.  V  ->  C  e.  Cat )
 
Theoremsetcid 13934 The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   =>    |-  ( ph  ->  (  .1.  `  X )  =  (  _I  |`  X ) )
 
Theoremsetcmon 13935 A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  M  =  (Mono `  C )   =>    |-  ( ph  ->  ( F  e.  ( X M Y )  <->  F : X -1-1-> Y ) )
 
Theoremsetcepi 13936 An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  E  =  (Epi `  C )   &    |-  ( ph  ->  2o  e.  U )   =>    |-  ( ph  ->  ( F  e.  ( X E Y )  <->  F : X -onto-> Y ) )
 
Theoremsetcsect 13937 A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  S  =  (Sect `  C )   =>    |-  ( ph  ->  ( F ( X S Y ) G  <->  ( F : X
 --> Y  /\  G : Y
 --> X  /\  ( G  o.  F )  =  (  _I  |`  X ) ) ) )
 
Theoremsetcinv 13938 An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  N  =  (Inv `  C )   =>    |-  ( ph  ->  ( F ( X N Y ) G  <->  ( F : X
 -1-1-onto-> Y  /\  G  =  `' F ) ) )
 
Theoremsetciso 13939 An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  I  =  ( 
 Iso  `  C )   =>    |-  ( ph  ->  ( F  e.  ( X I Y )  <->  F : X -1-1-onto-> Y ) )
 
Theoremresssetc 13940 The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the  SetCat `
 U categories for different  U are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  D  =  (
 SetCat `  V )   &    |-  ( ph  ->  U  e.  W )   &    |-  ( ph  ->  V  C_  U )   =>    |-  ( ph  ->  (
 (  Homf  `  ( Cs  V ) )  =  (  Homf  `  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
 
Theoremfuncsetcres2 13941 A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  D  =  (
 SetCat `  V )   &    |-  ( ph  ->  U  e.  W )   &    |-  ( ph  ->  V  C_  U )   =>    |-  ( ph  ->  ( E  Func  D )  C_  ( E  Func  C ) )
 
8.3.2  The category of categories
 
Syntaxccatc 13942 Extend class notation to include the category Cat.
 class CatCat
 
Definitiondf-catc 13943* Definition of the category Cat, which consists of all categories in the universe  u, with functors as the morphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- CatCat  =  ( u  e.  _V  |->  [_ ( u  i^i  Cat )  /  b ]_ { <. (
 Base `  ndx ) ,  b >. ,  <. (  Hom  ` 
 ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  Func  y )
 ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( b  X.  b ) ,  z  e.  b  |->  ( g  e.  ( ( 2nd `  v )  Func  z
 ) ,  f  e.  (  Func  `  v ) 
 |->  ( g  o.func  f )
 ) ) >. } )
 
Theoremcatcval 13944* Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  B  =  ( U  i^i  Cat ) )   &    |-  ( ph  ->  H  =  ( x  e.  B ,  y  e.  B  |->  ( x  Func  y ) ) )   &    |-  ( ph  ->  .x.  =  (
 v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v )  Func  z ) ,  f  e.  (  Func  `  v
 )  |->  ( g  o.func  f
 ) ) ) )   =>    |-  ( ph  ->  C  =  { <. ( Base `  ndx ) ,  B >. , 
 <. (  Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .x.  >. } )
 
Theoremcatcbas 13945 Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
 
Theoremcatchomfval 13946* Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  (  Hom  `  C )   =>    |-  ( ph  ->  H  =  ( x  e.  B ,  y  e.  B  |->  ( x  Func  y ) ) )
 
Theoremcatchom 13947 Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X H Y )  =  ( X  Func  Y ) )
 
Theoremcatccofval 13948* Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  .x.  =  (comp `  C )   =>    |-  ( ph  ->  .x. 
 =  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v
 )  Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) ) )
 
Theoremcatcco 13949 Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X  Func  Y ) )   &    |-  ( ph  ->  G  e.  ( Y  Func  Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  =  ( G  o.func 
 F ) )
 
Theoremcatccatid 13950* Lemma for catccat 13952. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   =>    |-  ( U  e.  V  ->  ( C  e.  Cat  /\  ( Id `  C )  =  ( x  e.  B  |->  (idfunc `  x ) ) ) )
 
Theoremcatcid 13951 The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  I  =  (idfunc `  X )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (  .1.  `  X )  =  I )
 
Theoremcatccat 13952 The category of categories is a category. (Clearly it cannot be an element of itself, hence it is "large" with respect to  U.) (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   =>    |-  ( U  e.  V  ->  C  e.  Cat )
 
Theoremresscatc 13953 The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat `  U categories for different  U are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  D  =  (CatCat `  V )   &    |-  ( ph  ->  U  e.  W )   &    |-  ( ph  ->  V 
 C_  U )   =>    |-  ( ph  ->  ( (  Homf  `  ( Cs  V ) )  =  (  Homf  `  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
 
Theoremcatcisolem 13954* Lemma for catciso 13955. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  R  =  (
 Base `  X )   &    |-  S  =  ( Base `  Y )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  (Inv `  C )   &    |-  H  =  ( x  e.  S ,  y  e.  S  |->  `' ( ( `' F `  x ) G ( `' F `  y ) ) )   &    |-  ( ph  ->  F ( ( X Full  Y )  i^i  ( X Faith  Y ) ) G )   &    |-  ( ph  ->  F : R
 -1-1-onto-> S )   =>    |-  ( ph  ->  <. F ,  G >. ( X I Y ) <. `' F ,  H >. )
 
Theoremcatciso 13955 A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  R  =  (
 Base `  X )   &    |-  S  =  ( Base `  Y )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  (  Iso  `  C )   =>    |-  ( ph  ->  ( F  e.  ( X I Y )  <->  ( F  e.  ( ( X Full  Y )  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) ) )
 
Theoremcatcoppccl 13956 The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  O  =  (oppCat `  X )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  O  e.  B )
 
Theoremcatcfuccl 13957 The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  Q  =  ( X FuncCat  Y )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  Q  e.  B )
 
8.4  Categorical constructions
 
8.4.1  Product of categories
 
Syntaxcxpc 13958 Extend class notation with the product of two categories.
 class  X.c
 
Syntaxc1stf 13959 Extend class notation with the first projection functor.
 class  1stF
 
Syntaxc2ndf 13960 Extend class notation with the second projection functor.
 class  2ndF
 
Syntaxcprf 13961 Extend class notation with the functor pairing operation.
 class ⟨,⟩F
 
Definitiondf-xpc 13962* Define the binary product of categories, which has objects for each pair of objects of the factors, and morphisms for each pair of morphisms of the factors. Composition is componentwise. (Contributed by Mario Carneiro, 10-Jan-2017.)
 |- 
 X.c 
 =  ( r  e. 
 _V ,  s  e. 
 _V  |->  [_ ( ( Base `  r )  X.  ( Base `  s ) ) 
 /  b ]_ [_ ( u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u ) (  Hom  `  r
 ) ( 1st `  v
 ) )  X.  (
 ( 2nd `  u )
 (  Hom  `  s ) ( 2nd `  v
 ) ) ) ) 
 /  h ]_ { <. (
 Base `  ndx ) ,  b >. ,  <. (  Hom  ` 
 ndx ) ,  h >. ,  <. (comp `  ndx ) ,  ( x  e.  ( b  X.  b
 ) ,  y  e.  b  |->  ( g  e.  ( ( 2nd `  x ) h y ) ,  f  e.  ( h `
  x )  |->  <.
 ( ( 1st `  g
 ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) ) >. (comp `  r ) ( 1st `  y ) ) ( 1st `  f )
 ) ,  ( ( 2nd `  g )
 ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
 >. (comp `  s )
 ( 2nd `  y )
 ) ( 2nd `  f
 ) ) >. ) )
 >. } )
 
Definitiondf-1stf 13963* Define the first projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 1stF  =  ( r  e.  Cat ,  s  e.  Cat  |->  [_ (
 ( Base `  r )  X.  ( Base `  s )
 )  /  b ]_ <. ( 1st  |`  b ) ,  ( x  e.  b ,  y  e.  b  |->  ( 1st  |`  ( x (  Hom  `  (
 r  X.c  s ) ) y ) ) ) >. )
 
Definitiondf-2ndf 13964* Define the second projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 2ndF  =  ( r  e.  Cat ,  s  e.  Cat  |->  [_ (
 ( Base `  r )  X.  ( Base `  s )
 )  /  b ]_ <. ( 2nd  |`  b ) ,  ( x  e.  b ,  y  e.  b  |->  ( 2nd  |`  ( x (  Hom  `  (
 r  X.c  s ) ) y ) ) ) >. )
 
Definitiondf-prf 13965* Define the pairing operation for functors (which takes two functors  F : C --> D and  G : C --> E and produces  ( F ⟨,⟩F  G ) : C --> ( D  X.c  E )). (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- ⟨,⟩F  =  ( f  e.  _V ,  g  e.  _V  |->  [_ dom  ( 1st `  f )  /  b ]_ <. ( x  e.  b  |->  <. ( ( 1st `  f ) `  x ) ,  (
 ( 1st `  g ) `  x ) >. ) ,  ( x  e.  b ,  y  e.  b  |->  ( h  e.  dom  ( x ( 2nd `  f
 ) y )  |->  <.
 ( ( x ( 2nd `  f )
 y ) `  h ) ,  ( ( x ( 2nd `  g
 ) y ) `  h ) >. ) )
 >. )
 
Theoremfnxpc 13966 The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
 |- 
 X.c 
 Fn  ( _V  X.  _V )
 
Theoremxpcval 13967* Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  X  =  (
 Base `  C )   &    |-  Y  =  ( Base `  D )   &    |-  H  =  (  Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  .x.  =  (comp `  C )   &    |-  .xb  =  (comp `  D )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  W )   &    |-  ( ph  ->  B  =  ( X  X.  Y ) )   &    |-  ( ph  ->  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v
 ) )  X.  (
 ( 2nd `  u ) J ( 2nd `  v
 ) ) ) ) )   &    |-  ( ph  ->  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `
  x )  |->  <.
 ( ( 1st `  g
 ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y ) ) ( 1st `  f
 ) ) ,  (
 ( 2nd `  g )
 ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
 >.  .xb  ( 2nd `  y
 ) ) ( 2nd `  f ) ) >. ) ) )   =>    |-  ( ph  ->  T  =  { <. ( Base `  ndx ) ,  B >. , 
 <. (  Hom  `  ndx ) ,  K >. , 
 <. (comp `  ndx ) ,  O >. } )
 
Theoremxpcbas 13968 Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  X  =  (
 Base `  C )   &    |-  Y  =  ( Base `  D )   =>    |-  ( X  X.  Y )  =  ( Base `  T )
 
Theoremxpchomfval 13969* Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  H  =  (  Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  K  =  ( 
 Hom  `  T )   =>    |-  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v )
 )  X.  ( ( 2nd `  u ) J ( 2nd `  v
 ) ) ) )
 
Theoremxpchom 13970 Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  H  =  (  Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  K  =  ( 
 Hom  `  T )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X K Y )  =  ( ( ( 1st `  X ) H ( 1st `  Y )
 )  X.  ( ( 2nd `  X ) J ( 2nd `  Y ) ) ) )
 
Theoremrelxpchom 13971 A hom-set in the binary product of categories is a relation. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  K  =  ( 
 Hom  `  T )   =>    |-  Rel  ( X K Y )
 
Theoremxpccofval 13972* Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  K  =  (  Hom  `  T )   &    |- 
 .x.  =  (comp `  C )   &    |-  .xb  =  (comp `  D )   &    |-  O  =  (comp `  T )   =>    |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `
  x )  |->  <.
 ( ( 1st `  g
 ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y ) ) ( 1st `  f
 ) ) ,  (
 ( 2nd `  g )
 ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
 >.  .xb  ( 2nd `  y
 ) ) ( 2nd `  f ) ) >. ) )
 
Theoremxpcco 13973 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  K  =  (  Hom  `  T )   &    |- 
 .x.  =  (comp `  C )   &    |-  .xb  =  (comp `  D )   &    |-  O  =  (comp `  T )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X K Y ) )   &    |-  ( ph  ->  G  e.  ( Y K Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  <. ( ( 1st `  G )
 ( <. ( 1st `  X ) ,  ( 1st `  Y ) >.  .x.  ( 1st `  Z ) ) ( 1st `  F ) ) ,  (
 ( 2nd `  G )
 ( <. ( 2nd `  X ) ,  ( 2nd `  Y ) >.  .xb  ( 2nd `  Z ) ) ( 2nd `  F ) ) >. )
 
Theoremxpcco1st 13974 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  K  =  (  Hom  `  T )   &    |-  O  =  (comp `  T )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X K Y ) )   &    |-  ( ph  ->  G  e.  ( Y K Z ) )   &    |-  .x. 
 =  (comp `  C )   =>    |-  ( ph  ->  ( 1st `  ( G (
 <. X ,  Y >. O Z ) F ) )  =  ( ( 1st `  G )
 ( <. ( 1st `  X ) ,  ( 1st `  Y ) >.  .x.  ( 1st `  Z ) ) ( 1st `  F ) ) )
 
Theoremxpcco2nd 13975 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  K  =  (  Hom  `  T )   &    |-  O  =  (comp `  T )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X K Y ) )   &    |-  ( ph  ->  G  e.  ( Y K Z ) )   &    |-  .x. 
 =  (comp `  D )   =>    |-  ( ph  ->  ( 2nd `  ( G (
 <. X ,  Y >. O Z ) F ) )  =  ( ( 2nd `  G )
 ( <. ( 2nd `  X ) ,  ( 2nd `  Y ) >.  .x.  ( 2nd `  Z ) ) ( 2nd `  F ) ) )
 
Theoremxpchom2 13976 Value of the set of morphisms in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  X  =  (
 Base `  C )   &    |-  Y  =  ( Base `  D )   &    |-  H  =  (  Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  ( ph  ->  M  e.  X )   &    |-  ( ph  ->  N  e.  Y )   &    |-  ( ph  ->  P  e.  X )   &    |-  ( ph  ->  Q  e.  Y )   &    |-  K  =  (  Hom  `  T )   =>    |-  ( ph  ->  ( <. M ,  N >. K
 <. P ,  Q >. )  =  ( ( M H P )  X.  ( N J Q ) ) )
 
Theoremxpcco2 13977 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  X  =  (
 Base `  C )   &    |-  Y  =  ( Base `  D )   &    |-  H  =  (  Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  ( ph  ->  M  e.  X )   &    |-  ( ph  ->  N  e.  Y )   &    |-  ( ph  ->  P  e.  X )   &    |-  ( ph  ->  Q  e.  Y )   &    |-  .x.  =  (comp `  C )   &    |-  .xb  =  (comp `  D )   &    |-  O  =  (comp `  T )   &    |-  ( ph  ->  R  e.  X )   &    |-  ( ph  ->  S  e.  Y )   &    |-  ( ph  ->  F  e.  ( M H P ) )   &    |-  ( ph  ->  G  e.  ( N J Q ) )   &    |-  ( ph  ->  K  e.  ( P H R ) )   &    |-  ( ph  ->  L  e.  ( Q J S ) )   =>    |-  ( ph  ->  (
 <. K ,  L >. (
 <. <. M ,  N >. ,  <. P ,  Q >.
 >. O <. R ,  S >. ) <. F ,  G >. )  =  <. ( K ( <. M ,  P >.  .x.  R ) F ) ,  ( L ( <. N ,  Q >. 
 .xb  S ) G )
 >. )
 
Theoremxpccatid 13978* The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  X  =  ( Base `  C )   &    |-  Y  =  (
 Base `  D )   &    |-  I  =  ( Id `  C )   &    |-  J  =  ( Id
 `  D )   =>    |-  ( ph  ->  ( T  e.  Cat  /\  ( Id `  T )  =  ( x  e.  X ,  y  e.  Y  |->  <. ( I `  x ) ,  ( J `  y ) >. ) ) )
 
Theoremxpcid 13979 The identity morphism in the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  X  =  ( Base `  C )   &    |-  Y  =  (
 Base `  D )   &    |-  I  =  ( Id `  C )   &    |-  J  =  ( Id
 `  D )   &    |-  .1.  =  ( Id `  T )   &    |-  ( ph  ->  R  e.  X )   &    |-  ( ph  ->  S  e.  Y )   =>    |-  ( ph  ->  (  .1.  `  <. R ,  S >. )  =  <. ( I `  R ) ,  ( J `  S ) >. )
 
Theoremxpccat 13980 The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  T  e.  Cat )
 
Theorem1stfval 13981* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  H  =  (  Hom  `  T )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  P  =  ( C  1stF  D )   =>    |-  ( ph  ->  P  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  ( x H y ) ) )
 >. )
 
Theorem1stf1 13982 Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  H  =  (  Hom  `  T )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  P  =  ( C  1stF  D )   &    |-  ( ph  ->  R  e.  B )   =>    |-  ( ph  ->  (
 ( 1st `  P ) `  R )  =  ( 1st `  R )
 )
 
Theorem1stf2 13983 Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  H  =  (  Hom  `  T )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  P  =  ( C  1stF  D )   &    |-  ( ph  ->  R  e.  B )   &    |-  ( ph  ->  S  e.  B )   =>    |-  ( ph  ->  ( R ( 2nd `  P ) S )  =  ( 1st  |`  ( R H S ) ) )
 
Theorem2ndfval 13984* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  H  =  (  Hom  `  T )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  Q  =  ( C  2ndF  D )   =>    |-  ( ph  ->  Q  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  ( x H y ) ) )
 >. )
 
Theorem2ndf1 13985 Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  H  =  (  Hom  `  T )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  Q  =  ( C  2ndF  D )   &    |-  ( ph  ->  R  e.  B )   =>    |-  ( ph  ->  (
 ( 1st `  Q ) `  R )  =  ( 2nd `  R )
 )
 
Theorem2ndf2 13986 Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  B  =  (
 Base `  T )   &    |-  H  =  (  Hom  `  T )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  Q  =  ( C  2ndF  D )   &    |-  ( ph  ->  R  e.  B )   &    |-  ( ph  ->  S  e.  B )   =>    |-  ( ph  ->  ( R ( 2nd `  Q ) S )  =  ( 2nd  |`  ( R H S ) ) )
 
Theorem1stfcl 13987 The first projection functor is a functor onto the left argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  P  =  ( C 
 1stF  D )   =>    |-  ( ph  ->  P  e.  ( T  Func  C ) )
 
Theorem2ndfcl 13988 The second projection functor is a functor onto the right argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  T  =  ( C  X.c  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  Q  =  ( C 
 2ndF  D )   =>    |-  ( ph  ->  Q  e.  ( T  Func  D ) )
 
Theoremprfval 13989* Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  P  =  ( F ⟨,⟩F  G )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  E ) )   =>    |-  ( ph  ->  P  =  <. ( x  e.  B  |->  <. ( ( 1st `  F ) `  x ) ,  ( ( 1st `  G ) `  x ) >. ) ,  ( x  e.  B ,  y  e.  B  |->  ( h  e.  ( x H y )  |->  <.
 ( ( x ( 2nd `  F )
 y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `  h ) >. ) )
 >. )
 
Theoremprf1 13990 Value of the pairing functor on objects. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  P  =  ( F ⟨,⟩F  G )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  E ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( ( 1st `  P ) `  X )  = 
 <. ( ( 1st `  F ) `  X ) ,  ( ( 1st `  G ) `  X ) >. )
 
Theoremprf2fval 13991* Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  P  =  ( F ⟨,⟩F  G )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  E ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X ( 2nd `  P ) Y )  =  ( h  e.  ( X H Y )  |->  <.
 ( ( X ( 2nd `  F ) Y ) `  h ) ,  ( ( X ( 2nd `  G ) Y ) `  h ) >. ) )
 
Theoremprf2 13992 Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  P  =  ( F ⟨,⟩F  G )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  E ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  K  e.  ( X H Y ) )   =>    |-  ( ph  ->  (
 ( X ( 2nd `  P ) Y ) `
  K )  = 
 <. ( ( X ( 2nd `  F ) Y ) `  K ) ,  ( ( X ( 2nd `  G ) Y ) `  K ) >. )
 
Theoremprfcl 13993 The pairing of functors  F : C --> D and  G : C --> D is a functor  <. F ,  G >. : C --> ( D  X.  E ). (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  P  =  ( F ⟨,⟩F  G )   &    |-  T  =  ( D  X.c  E )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  E ) )   =>    |-  ( ph  ->  P  e.  ( C  Func  T ) )
 
Theoremprf1st 13994 Cancellation of pairing with first projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  P  =  ( F ⟨,⟩F  G )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  E ) )   =>    |-  ( ph  ->  (
 ( D  1stF  E )  o.func  P )  =  F )
 
Theoremprf2nd 13995 Cancellation of pairing with second projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  P  =  ( F ⟨,⟩F  G )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  E ) )   =>    |-  ( ph  ->  (
 ( D  2ndF  E )  o.func  P )  =  G )
 
Theorem1st2ndprf 13996 Break a functor into a product category into first and second projections. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  T  =  ( D  X.c  E )   &    |-  ( ph  ->  F  e.  ( C  Func  T ) )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   =>    |-  ( ph  ->  F  =  ( ( ( D 
 1stF  E )  o.func 
 F ) ⟨,⟩F  ( ( D  2ndF  E )  o.func 
 F ) ) )
 
Theoremcatcxpccl 13997 The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  T  =  ( X  X.c  Y )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  T  e.  B )
 
Theoremxpcpropd 13998 If two categories have the same set of objects, morphisms, and compositions, then they have the same product category. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  A )  =  ( 
 Homf  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  (  Homf  `  C )  =  (  Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  ( A  X.c  C )  =  ( B  X.c  D ) )
 
8.4.2  Functor evaluation
 
Syntaxcevlf 13999 Extend class notation with the evaluation functor.
 class evalF
 
Syntaxccurf 14000 Extend class notation with the currying of a functor.
 class curryF
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