HomeHome Metamath Proof Explorer
Theorem List (p. 160 of 410)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26627)
  Hilbert Space Explorer  Hilbert Space Explorer
(26628-28150)
  Users' Mathboxes  Users' Mathboxes
(28151-40909)
 

Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
8.1.9  Natural transformations and the functor category
 
Syntaxcnat 15901 Extend class notation to include the collection of natural transformations.
 class Nat
 
Syntaxcfuc 15902 Extend class notation to include the functor category.
 class FuncCat
 
Definitiondf-nat 15903* Definition of a natural transformation between two functors. A natural transformation  A : F --> G is a collection of arrows  A ( x ) : F ( x ) --> G ( x ), such that  A ( y )  o.  F ( h )  =  G ( h )  o.  A ( x ) for each morphism  h : x --> y. Definition 6.1 in [Adamek] p. 83, and definition in [Lang] p. 65. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |- Nat 
 =  ( t  e. 
 Cat ,  u  e.  Cat  |->  ( f  e.  (
 t  Func  u ) ,  g  e.  ( t 
 Func  u )  |->  [_ ( 1st `  f )  /  r ]_ [_ ( 1st `  g )  /  s ]_ { a  e.  X_ x  e.  ( Base `  t ) ( ( r `  x ) ( Hom  `  u ) ( s `  x ) )  | 
 A. x  e.  ( Base `  t ) A. y  e.  ( Base `  t ) A. h  e.  ( x ( Hom  `  t ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y ) >. (comp `  u ) ( s `
  y ) ) ( ( x ( 2nd `  f )
 y ) `  h ) )  =  (
 ( ( x ( 2nd `  g )
 y ) `  h ) ( <. ( r `
  x ) ,  ( s `  x ) >. (comp `  u ) ( s `  y ) ) ( a `  x ) ) } ) )
 
Definitiondf-fuc 15904* Definition of the category of functors between two fixed categories, with the objects being functors and the morphisms being natural transformations. Definition 6.15 in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |- FuncCat  =  ( t  e.  Cat ,  u  e.  Cat  |->  { <. (
 Base `  ndx ) ,  ( t  Func  u ) >. ,  <. ( Hom  `  ndx ) ,  (
 t Nat  u ) >. , 
 <. (comp `  ndx ) ,  ( v  e.  (
 ( t  Func  u )  X.  ( t  Func  u ) ) ,  h  e.  ( t  Func  u )  |->  [_ ( 1st `  v
 )  /  f ]_ [_ ( 2nd `  v
 )  /  g ]_ ( b  e.  (
 g ( t Nat  u ) h ) ,  a  e.  ( f ( t Nat 
 u ) g ) 
 |->  ( x  e.  ( Base `  t )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x ) ,  (
 ( 1st `  g ) `  x ) >. (comp `  u ) ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) ) )
 >. } )
 
Theoremfnfuc 15905 The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |- FuncCat  Fn  ( Cat  X.  Cat )
 
Theoremnatfval 15906* Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  J  =  ( Hom  `  D )   &    |-  .x.  =  (comp `  D )   =>    |-  N  =  ( f  e.  ( C  Func  D ) ,  g  e.  ( C  Func  D ) 
 |->  [_ ( 1st `  f
 )  /  r ]_ [_ ( 1st `  g
 )  /  s ]_ { a  e.  X_ x  e.  B  ( ( r `
  x ) J ( s `  x ) )  |  A. x  e.  B  A. y  e.  B  A. h  e.  ( x H y ) ( ( a `
  y ) (
 <. ( r `  x ) ,  ( r `  y ) >.  .x.  (
 s `  y )
 ) ( ( x ( 2nd `  f
 ) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
 ) y ) `  h ) ( <. ( r `  x ) ,  ( s `  x ) >.  .x.  (
 s `  y )
 ) ( a `  x ) ) }
 )
 
Theoremisnat 15907* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  J  =  ( Hom  `  D )   &    |-  .x.  =  (comp `  D )   &    |-  ( ph  ->  F ( C  Func  D ) G )   &    |-  ( ph  ->  K ( C  Func  D ) L )   =>    |-  ( ph  ->  ( A  e.  ( <. F ,  G >. N <. K ,  L >. )  <->  ( A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) )  /\  A. x  e.  B  A. y  e.  B  A. h  e.  ( x H y ) ( ( A `
  y ) (
 <. ( F `  x ) ,  ( F `  y ) >.  .x.  ( K `  y ) ) ( ( x G y ) `  h ) )  =  (
 ( ( x L y ) `  h ) ( <. ( F `
  x ) ,  ( K `  x ) >.  .x.  ( K `  y ) ) ( A `  x ) ) ) ) )
 
Theoremisnat2 15908* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  J  =  ( Hom  `  D )   &    |-  .x.  =  (comp `  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   =>    |-  ( ph  ->  ( A  e.  ( F N G )  <->  ( A  e.  X_ x  e.  B  ( ( ( 1st `  F ) `  x ) J ( ( 1st `  G ) `  x ) ) 
 /\  A. x  e.  B  A. y  e.  B  A. h  e.  ( x H y ) ( ( A `  y
 ) ( <. ( ( 1st `  F ) `  x ) ,  (
 ( 1st `  F ) `  y ) >.  .x.  (
 ( 1st `  G ) `  y ) ) ( ( x ( 2nd `  F ) y ) `
  h ) )  =  ( ( ( x ( 2nd `  G ) y ) `  h ) ( <. ( ( 1st `  F ) `  x ) ,  ( ( 1st `  G ) `  x ) >.  .x.  ( ( 1st `  G ) `  y ) ) ( A `  x ) ) ) ) )
 
Theoremnatffn 15909 The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   =>    |-  N  Fn  ( ( C  Func  D )  X.  ( C  Func  D ) )
 
Theoremnatrcl 15910 Reverse closure for a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   =>    |-  ( A  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
 
Theoremnat1st2nd 15911 Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  A  e.  ( F N G ) )   =>    |-  ( ph  ->  A  e.  ( <. ( 1st `  F ) ,  ( 2nd `  F ) >. N
 <. ( 1st `  G ) ,  ( 2nd `  G ) >. ) )
 
Theoremnatixp 15912* A natural transformation is a function from the objects of  C to homomorphisms from  F ( x ) to  G ( x ). (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  A  e.  ( <. F ,  G >. N <. K ,  L >. ) )   &    |-  B  =  ( Base `  C )   &    |-  J  =  ( Hom  `  D )   =>    |-  ( ph  ->  A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) ) )
 
Theoremnatcl 15913 A component of a natural transformation is a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  A  e.  ( <. F ,  G >. N <. K ,  L >. ) )   &    |-  B  =  ( Base `  C )   &    |-  J  =  ( Hom  `  D )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( A `  X )  e.  ( ( F `  X ) J ( K `  X ) ) )
 
Theoremnatfn 15914 A natural transformation is a function on the objects of  C. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  A  e.  ( <. F ,  G >. N <. K ,  L >. ) )   &    |-  B  =  ( Base `  C )   =>    |-  ( ph  ->  A  Fn  B )
 
Theoremnati 15915 Naturality property of a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  A  e.  ( <. F ,  G >. N <. K ,  L >. ) )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |- 
 .x.  =  (comp `  D )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( ( A `  Y ) (
 <. ( F `  X ) ,  ( F `  Y ) >.  .x.  ( K `  Y ) ) ( ( X G Y ) `  R ) )  =  (
 ( ( X L Y ) `  R ) ( <. ( F `
  X ) ,  ( K `  X ) >.  .x.  ( K `  Y ) ) ( A `  X ) ) )
 
Theoremwunnat 15916 A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  C  e.  U )   &    |-  ( ph  ->  D  e.  U )   =>    |-  ( ph  ->  ( C Nat  D )  e.  U )
 
Theoremcatstr 15917 A category structure is a structure. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- 
 { <. ( Base `  ndx ) ,  U >. , 
 <. ( Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .x.  >. } Struct  <. 1 , ; 1
 5 >.
 
Theoremfucval 15918* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  ( C  Func  D )   &    |-  N  =  ( C Nat  D )   &    |-  A  =  ( Base `  C )   &    |-  .x.  =  (comp `  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  .xb  =  ( v  e.  ( B  X.  B ) ,  h  e.  B  |->  [_ ( 1st `  v )  /  f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g )  |->  ( x  e.  A  |->  ( ( b `
  x ) (
 <. ( ( 1st `  f
 ) `  x ) ,  ( ( 1st `  g
 ) `  x ) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `
  x ) ) ) ) ) )   =>    |-  ( ph  ->  Q  =  { <. ( Base `  ndx ) ,  B >. , 
 <. ( Hom  `  ndx ) ,  N >. , 
 <. (comp `  ndx ) , 
 .xb  >. } )
 
Theoremfuccofval 15919* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  ( C  Func  D )   &    |-  N  =  ( C Nat  D )   &    |-  A  =  ( Base `  C )   &    |-  .x.  =  (comp `  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  .xb  =  (comp `  Q )   =>    |-  ( ph  ->  .xb  =  ( v  e.  ( B  X.  B ) ,  h  e.  B  |->  [_ ( 1st `  v )  /  f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g )  |->  ( x  e.  A  |->  ( ( b `
  x ) (
 <. ( ( 1st `  f
 ) `  x ) ,  ( ( 1st `  g
 ) `  x ) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `
  x ) ) ) ) ) )
 
Theoremfucbas 15920 The objects of the functor category are functors from  C to  D. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 12-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   =>    |-  ( C  Func  D )  =  ( Base `  Q )
 
Theoremfuchom 15921 The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   =>    |-  N  =  ( Hom  `  Q )
 
Theoremfucco 15922* Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  A  =  ( Base `  C )   &    |-  .x.  =  (comp `  D )   &    |-  .xb  =  (comp `  Q )   &    |-  ( ph  ->  R  e.  ( F N G ) )   &    |-  ( ph  ->  S  e.  ( G N H ) )   =>    |-  ( ph  ->  ( S ( <. F ,  G >.  .xb  H ) R )  =  ( x  e.  A  |->  ( ( S `  x ) ( <. ( ( 1st `  F ) `  x ) ,  ( ( 1st `  G ) `  x ) >.  .x.  (
 ( 1st `  H ) `  x ) ) ( R `  x ) ) ) )
 
Theoremfuccoval 15923 Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  A  =  ( Base `  C )   &    |-  .x.  =  (comp `  D )   &    |-  .xb  =  (comp `  Q )   &    |-  ( ph  ->  R  e.  ( F N G ) )   &    |-  ( ph  ->  S  e.  ( G N H ) )   &    |-  ( ph  ->  X  e.  A )   =>    |-  ( ph  ->  ( ( S ( <. F ,  G >.  .xb  H ) R ) `  X )  =  ( ( S `  X ) (
 <. ( ( 1st `  F ) `  X ) ,  ( ( 1st `  G ) `  X ) >.  .x.  ( ( 1st `  H ) `  X ) ) ( R `  X ) ) )
 
Theoremfuccocl 15924 The composition of two natural transformations is a natural transformation. Remark 6.14(a) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  .xb  =  (comp `  Q )   &    |-  ( ph  ->  R  e.  ( F N G ) )   &    |-  ( ph  ->  S  e.  ( G N H ) )   =>    |-  ( ph  ->  ( S ( <. F ,  G >.  .xb  H ) R )  e.  ( F N H ) )
 
Theoremfucidcl 15925 The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  .1.  =  ( Id `  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   =>    |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  e.  ( F N F ) )
 
Theoremfuclid 15926 Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  .xb  =  (comp `  Q )   &    |-  .1.  =  ( Id `  D )   &    |-  ( ph  ->  R  e.  ( F N G ) )   =>    |-  ( ph  ->  (
 (  .1.  o.  ( 1st `  G ) ) ( <. F ,  G >. 
 .xb  G ) R )  =  R )
 
Theoremfucrid 15927 Right identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  .xb  =  (comp `  Q )   &    |-  .1.  =  ( Id `  D )   &    |-  ( ph  ->  R  e.  ( F N G ) )   =>    |-  ( ph  ->  ( R ( <. F ,  F >.  .xb  G ) (  .1.  o.  ( 1st `  F ) ) )  =  R )
 
Theoremfucass 15928 Associativity of natural transformation composition. Remark 6.14(b) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  .xb  =  (comp `  Q )   &    |-  ( ph  ->  R  e.  ( F N G ) )   &    |-  ( ph  ->  S  e.  ( G N H ) )   &    |-  ( ph  ->  T  e.  ( H N K ) )   =>    |-  ( ph  ->  ( ( T ( <. G ,  H >.  .xb  K ) S ) ( <. F ,  G >.  .xb  K ) R )  =  ( T ( <. F ,  H >.  .xb  K ) ( S ( <. F ,  G >.  .xb  H ) R ) ) )
 
Theoremfuccatid 15929* The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |- 
 .1.  =  ( Id `  D )   =>    |-  ( ph  ->  ( Q  e.  Cat  /\  ( Id `  Q )  =  ( f  e.  ( C  Func  D )  |->  (  .1.  o.  ( 1st `  f ) ) ) ) )
 
Theoremfuccat 15930 The functor category is a category. Remark 6.16 in [Adamek] p. 88. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  Q  e.  Cat )
 
Theoremfucid 15931 The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  I  =  ( Id `  Q )   &    |-  .1.  =  ( Id `  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   =>    |-  ( ph  ->  ( I `  F )  =  (  .1.  o.  ( 1st `  F ) ) )
 
Theoremfucsect 15932* Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  (
 Base `  C )   &    |-  N  =  ( C Nat  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  S  =  (Sect `  Q )   &    |-  T  =  (Sect `  D )   =>    |-  ( ph  ->  ( U ( F S G ) V  <->  ( U  e.  ( F N G ) 
 /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x ) T ( ( 1st `  G ) `  x ) ) ( V `
  x ) ) ) )
 
Theoremfucinv 15933* Two natural transformations are inverses of each other iff all the components are inverse. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  (
 Base `  C )   &    |-  N  =  ( C Nat  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  I  =  (Inv `  Q )   &    |-  J  =  (Inv `  D )   =>    |-  ( ph  ->  ( U ( F I G ) V  <->  ( U  e.  ( F N G ) 
 /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x ) J ( ( 1st `  G ) `  x ) ) ( V `
  x ) ) ) )
 
Theoreminvfuc 15934* If  V (
x ) is an inverse to  U ( x ) for each  x, and  U is a natural transformation, then  V is also a natural transformation, and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  (
 Base `  C )   &    |-  N  =  ( C Nat  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  I  =  (Inv `  Q )   &    |-  J  =  (Inv `  D )   &    |-  ( ph  ->  U  e.  ( F N G ) )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( U `  x ) ( ( ( 1st `  F ) `  x ) J ( ( 1st `  G ) `  x ) ) X )   =>    |-  ( ph  ->  U ( F I G ) ( x  e.  B  |->  X ) )
 
Theoremfuciso 15935* A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  (
 Base `  C )   &    |-  N  =  ( C Nat  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  I  =  ( 
 Iso  `  Q )   &    |-  J  =  (  Iso  `  D )   =>    |-  ( ph  ->  ( A  e.  ( F I G )  <->  ( A  e.  ( F N G ) 
 /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `  x ) J ( ( 1st `  G ) `  x ) ) ) ) )
 
Theoremnatpropd 15936 If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  A  e.  Cat )   &    |-  ( ph  ->  B  e.  Cat )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  ( A Nat  C )  =  ( B Nat  D ) )
 
Theoremfucpropd 15937 If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  A  e.  Cat )   &    |-  ( ph  ->  B  e.  Cat )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  ( A FuncCat  C )  =  ( B FuncCat  D ) )
 
8.1.10  Initial, terminal and zero objects of a category
 
Syntaxcinito 15938 Extend class notation with the class of initial objects of a category.
 class InitO
 
Syntaxctermo 15939 Extend class notation with the class of terminal objects of a category.
 class TermO
 
Syntaxczeroo 15940 Extend class notation with the class of zero objects of a category.
 class ZeroO
 
Definitiondf-inito 15941* An object A is said to be an initial object provided that for each object B there is exactly one morphism from A to B. Definition 7.1 in [Adamek] p. 101, or definition in [Lang] p. 57 (called "a universally repelling object" there). (Contributed by AV, 3-Apr-2020.)
 |- InitO  =  ( c  e.  Cat  |->  { a  e.  ( Base `  c )  |  A. b  e.  ( Base `  c ) E! h  h  e.  ( a
 ( Hom  `  c ) b ) } )
 
Definitiondf-termo 15942* An object A is called a terminal object provided that for each object B there is exactly one morphism from B to A. Definition 7.4 in [Adamek] p. 102, or definition in [Lang] p. 57 (called "a universally attracting object" there). (Contributed by AV, 3-Apr-2020.)
 |- TermO  =  ( c  e.  Cat  |->  { a  e.  ( Base `  c )  |  A. b  e.  ( Base `  c ) E! h  h  e.  ( b
 ( Hom  `  c ) a ) } )
 
Definitiondf-zeroo 15943 An object A is called a zero object provided that it is both an initial object and a terminal object. Definition 7.7 of [Adamek] p. 103. (Contributed by AV, 3-Apr-2020.)
 |- ZeroO  =  ( c  e.  Cat  |->  ( (InitO `  c )  i^i  (TermO `  c )
 ) )
 
Theoreminitorcl 15944 Reverse closure for an initial object: If a class has an initial object, the class is a category. (Contributed by AV, 4-Apr-2020.)
 |-  ( I  e.  (InitO `  C )  ->  C  e.  Cat )
 
Theoremtermorcl 15945 Reverse closure for a terminal object: If a class has a terminal object, the class is a category. (Contributed by AV, 4-Apr-2020.)
 |-  ( T  e.  (TermO `  C )  ->  C  e.  Cat )
 
Theoremzeroorcl 15946 Reverse closure for a zero object: If a class has a zero object, the class is a category. (Contributed by AV, 4-Apr-2020.)
 |-  ( Z  e.  (ZeroO `  C )  ->  C  e.  Cat )
 
Theoreminitoval 15947* The value of the initial object function, i.e. the set of all initial objects of a category. (Contributed by AV, 3-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( ph  ->  (InitO `  C )  =  {
 a  e.  B  |  A. b  e.  B  E! h  h  e.  ( a H b ) } )
 
Theoremtermoval 15948* The value of the terminal object function, i.e. the set of all terminal objects of a category. (Contributed by AV, 3-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( ph  ->  (TermO `  C )  =  {
 a  e.  B  |  A. b  e.  B  E! h  h  e.  ( b H a ) } )
 
Theoremzerooval 15949 The value of the zero object function, i.e. the set of all zero objects of a category. (Contributed by AV, 3-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( ph  ->  (ZeroO `  C )  =  ( (InitO `  C )  i^i  (TermO `  C )
 ) )
 
Theoremisinito 15950* The predicate "is an initial object" of a category. (Contributed by AV, 3-Apr-2020.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  I  e.  B )   =>    |-  ( ph  ->  ( I  e.  (InitO `  C ) 
 <-> 
 A. b  e.  B  E! h  h  e.  ( I H b ) ) )
 
Theoremistermo 15951* The predicate "is a terminal object" of a category. (Contributed by AV, 3-Apr-2020.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  I  e.  B )   =>    |-  ( ph  ->  ( I  e.  (TermO `  C ) 
 <-> 
 A. b  e.  B  E! h  h  e.  ( b H I ) ) )
 
Theoremiszeroo 15952 The predicate "is a zero object" of a category. (Contributed by AV, 3-Apr-2020.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  I  e.  B )   =>    |-  ( ph  ->  ( I  e.  (ZeroO `  C ) 
 <->  ( I  e.  (InitO `  C )  /\  I  e.  (TermO `  C )
 ) ) )
 
Theoremisinitoi 15953* Implication of a class being an initial object. (Contributed by AV, 6-Apr-2020.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  C  e.  Cat )   =>    |-  ( ( ph  /\  O  e.  (InitO `  C )
 )  ->  ( O  e.  B  /\  A. b  e.  B  E! h  h  e.  ( O H b ) ) )
 
Theoremistermoi 15954* Implication of a class being a terminal object. (Contributed by AV, 18-Apr-2020.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  C  e.  Cat )   =>    |-  ( ( ph  /\  O  e.  (TermO `  C )
 )  ->  ( O  e.  B  /\  A. b  e.  B  E! h  h  e.  ( b H O ) ) )
 
Theoreminitoid 15955 For an initial object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 6-Apr-2020.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  C  e.  Cat )   =>    |-  ( ( ph  /\  O  e.  (InitO `  C )
 )  ->  ( O H O )  =  {
 ( ( Id `  C ) `  O ) } )
 
Theoremtermoid 15956 For a terminal object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 18-Apr-2020.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  C  e.  Cat )   =>    |-  ( ( ph  /\  O  e.  (TermO `  C )
 )  ->  ( O H O )  =  {
 ( ( Id `  C ) `  O ) } )
 
Theoreminitoo 15957 An initial object is an object. (Contributed by AV, 14-Apr-2020.)
 |-  ( C  e.  Cat  ->  ( O  e.  (InitO `  C )  ->  O  e.  ( Base `  C )
 ) )
 
Theoremtermoo 15958 A terminal object is an object. (Contributed by AV, 18-Apr-2020.)
 |-  ( C  e.  Cat  ->  ( O  e.  (TermO `  C )  ->  O  e.  ( Base `  C )
 ) )
 
Theoremiszeroi 15959 Implication of a class being a zero object. (Contributed by AV, 18-Apr-2020.)
 |-  ( ( C  e.  Cat  /\  O  e.  (ZeroO `  C ) )  ->  ( O  e.  ( Base `  C )  /\  ( O  e.  (InitO `  C )  /\  O  e.  (TermO `  C )
 ) ) )
 
Theorem2initoinv 15960 Morphisms between two initial objects are inverses. (Contributed by AV, 14-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  A  e.  (InitO `  C ) )   &    |-  ( ph  ->  B  e.  (InitO `  C ) )   =>    |-  ( ( ph  /\  G  e.  ( B ( Hom  `  C ) A ) 
 /\  F  e.  ( A ( Hom  `  C ) B ) )  ->  F ( A (Inv `  C ) B ) G )
 
Theoreminitoeu1 15961* Initial objects are essentially unique (strong form), i.e. there is a unique isomorphism between two initial objects, see statement in [Lang] p. 58 ("... if P, P' are two universal objects [...] then there exists a unique isomorphism between them.". (Proposed by BJ, 14-Apr-2020.) (Contributed by AV, 14-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  A  e.  (InitO `  C ) )   &    |-  ( ph  ->  B  e.  (InitO `  C ) )   =>    |-  ( ph  ->  E! f  f  e.  ( A (  Iso  `  C ) B ) )
 
Theoreminitoeu1w 15962 Initial objects are essentially unique (weak form), i.e. if A and B are initial objects, then A and B are isomorphic. Proposition 7.3 (1) of [Adamek] p. 102. (Contributed by AV, 6-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  A  e.  (InitO `  C ) )   &    |-  ( ph  ->  B  e.  (InitO `  C ) )   =>    |-  ( ph  ->  A (  ~=c𝑐  `  C ) B )
 
Theoreminitoeu2lem0 15963 Lemma 0 for initoeu2 15966. (Contributed by AV, 9-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  A  e.  (InitO `  C ) )   &    |-  X  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  I  =  (  Iso  `  C )   &    |-  .o.  =  (comp `  C )   =>    |-  ( ( ( ph  /\  ( A  e.  X  /\  B  e.  X  /\  D  e.  X )
 )  /\  ( K  e.  ( B I A )  /\  F  e.  ( A H D ) 
 /\  G  e.  ( B H D ) ) 
 /\  ( ( F ( <. B ,  A >.  .o.  D ) K ) ( <. A ,  B >.  .o.  D )
 ( ( B (Inv `  C ) A ) `
  K ) )  =  ( G (
 <. A ,  B >.  .o. 
 D ) ( ( B (Inv `  C ) A ) `  K ) ) )  ->  G  =  ( F ( <. B ,  A >.  .o.  D ) K ) )
 
Theoreminitoeu2lem1 15964* Lemma 1 for initoeu2 15966. (Contributed by AV, 9-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  A  e.  (InitO `  C ) )   &    |-  X  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  I  =  (  Iso  `  C )   &    |-  .o.  =  (comp `  C )   =>    |-  ( ( ph  /\  ( A  e.  X  /\  B  e.  X  /\  D  e.  X )  /\  ( K  e.  ( B I A )  /\  ( F ( <. B ,  A >.  .o.  D ) K )  e.  ( B H D ) ) )  ->  ( ( E! f  f  e.  ( A H D ) 
 /\  F  e.  ( A H D )  /\  G  e.  ( B H D ) )  ->  G  =  ( F ( <. B ,  A >.  .o.  D ) K ) ) )
 
Theoreminitoeu2lem2 15965* Lemma 2 for initoeu2 15966. (Contributed by AV, 10-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  A  e.  (InitO `  C ) )   &    |-  X  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  I  =  (  Iso  `  C )   &    |-  .o.  =  (comp `  C )   =>    |-  ( ( ph  /\  ( A  e.  X  /\  B  e.  X  /\  D  e.  X )  /\  ( K  e.  ( B I A )  /\  F  e.  ( A H D )  /\  ( F ( <. B ,  A >.  .o.  D ) K )  e.  ( B H D ) ) )  ->  ( E! f  f  e.  ( A H D )  ->  E! g  g  e.  ( B H D ) ) )
 
Theoreminitoeu2 15966 Initial objects are essentially unique, if A is an initial object, then so is every object that is isomorphic to A. Proposition 7.3 (2) in [Adamek] p. 102. (Contributed by AV, 10-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  A  e.  (InitO `  C ) )   &    |-  ( ph  ->  A (  ~=c𝑐  `  C ) B )   =>    |-  ( ph  ->  B  e.  (InitO `  C ) )
 
Theorem2termoinv 15967 Morphisms between two terminal objects are inverses. (Contributed by AV, 18-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  A  e.  (TermO `  C ) )   &    |-  ( ph  ->  B  e.  (TermO `  C ) )   =>    |-  ( ( ph  /\  G  e.  ( B ( Hom  `  C ) A ) 
 /\  F  e.  ( A ( Hom  `  C ) B ) )  ->  F ( A (Inv `  C ) B ) G )
 
Theoremtermoeu1 15968* Terminal objects are essentially unique (strong form), i.e. there is a unique isomorphism between two terminal objects, see statement in [Lang] p. 58 ("... if P, P' are two universal objects [...] then there exists a unique isomorphism between them.". (Proposed by BJ, 14-Apr-2020.) (Contributed by AV, 18-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  A  e.  (TermO `  C ) )   &    |-  ( ph  ->  B  e.  (TermO `  C ) )   =>    |-  ( ph  ->  E! f  f  e.  ( A (  Iso  `  C ) B ) )
 
Theoremtermoeu1w 15969 Terminal objects are essentially unique (weak form), i.e. if A and B are terminal objects, then A and B are isomorphic. Proposition 7.6 of [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
 |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  A  e.  (TermO `  C ) )   &    |-  ( ph  ->  B  e.  (TermO `  C ) )   =>    |-  ( ph  ->  A (  ~=c𝑐  `  C ) B )
 
8.2  Arrows (disjointified hom-sets)
 
Syntaxcdoma 15970 Extend class notation to include the domain extractor for an arrow.
 class domA
 
Syntaxccoda 15971 Extend class notation to include the codomain extractor for an arrow.
 class coda
 
Syntaxcarw 15972 Extend class notation to include the collection of all arrows of a category.
 class Nat
 
Syntaxchoma 15973 Extend class notation to include the set of all arrows with a specific domain and codomain.
 class Homa
 
Definitiondf-doma 15974 Definition of the domain extractor for an arrow. (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
 |- domA  =  ( 1st  o.  1st )
 
Definitiondf-coda 15975 Definition of the codomain extractor for an arrow. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
 |- coda  =  ( 2nd  o.  1st )
 
Definitiondf-homa 15976* Definition of the hom-set extractor for arrows, which tags the morphisms of the underlying hom-set with domain and codomain, which can then be extracted using df-doma 15974 and df-coda 15975. (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
 |- Homa  =  ( c  e.  Cat  |->  ( x  e.  (
 ( Base `  c )  X.  ( Base `  c )
 )  |->  ( { x }  X.  ( ( Hom  `  c ) `  x ) ) ) )
 
Definitiondf-arw 15977 Definition of the set of arrows of a category. We will use the term "arrow" to denote a morphism tagged with its domain and codomain, as opposed to  Hom, which allows hom-sets for distinct objects to overlap. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- Nat 
 =  ( c  e. 
 Cat  |->  U. ran  (Homa `  c
 ) )
 
Theoremhomarcl 15978 Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   =>    |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
 
Theoremhomafval 15979* Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  J  =  ( Hom  `  C )   =>    |-  ( ph  ->  H  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x ) ) ) )
 
Theoremhomaf 15980 Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   =>    |-  ( ph  ->  H : ( B  X.  B ) --> ~P (
 ( B  X.  B )  X.  _V ) )
 
Theoremhomaval 15981 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  J  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X H Y )  =  ( { <. X ,  Y >. }  X.  ( X J Y ) ) )
 
Theoremelhoma 15982 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  J  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( Z ( X H Y ) F  <->  ( Z  =  <. X ,  Y >.  /\  F  e.  ( X J Y ) ) ) )
 
Theoremelhomai 15983 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  J  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X J Y ) )   =>    |-  ( ph  ->  <. X ,  Y >. ( X H Y ) F )
 
Theoremelhomai2 15984 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  J  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X J Y ) )   =>    |-  ( ph  ->  <. X ,  Y ,  F >.  e.  ( X H Y ) )
 
Theoremhomarcl2 15985 Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |-  B  =  ( Base `  C )   =>    |-  ( F  e.  ( X H Y )  ->  ( X  e.  B  /\  Y  e.  B ) )
 
Theoremhomarel 15986 An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   =>    |- 
 Rel  ( X H Y )
 
Theoremhoma1 15987 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   =>    |-  ( Z ( X H Y ) F 
 ->  Z  =  <. X ,  Y >. )
 
Theoremhomahom2 15988 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |-  J  =  ( Hom  `  C )   =>    |-  ( Z ( X H Y ) F 
 ->  F  e.  ( X J Y ) )
 
Theoremhomahom 15989 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |-  J  =  ( Hom  `  C )   =>    |-  ( F  e.  ( X H Y )  ->  ( 2nd `  F )  e.  ( X J Y ) )
 
Theoremhomadm 15990 The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   =>    |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  X )
 
Theoremhomacd 15991 The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   =>    |-  ( F  e.  ( X H Y )  ->  (coda `  F )  =  Y )
 
Theoremhomadmcd 15992 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   =>    |-  ( F  e.  ( X H Y )  ->  F  =  <. X ,  Y ,  ( 2nd `  F ) >. )
 
Theoremarwval 15993 The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  A  =  (Nat `  C )   &    |-  H  =  (Homa `  C )   =>    |-  A  =  U. ran  H
 
Theoremarwrcl 15994 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  A  =  (Nat `  C )   =>    |-  ( F  e.  A  ->  C  e.  Cat )
 
Theoremarwhoma 15995 An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  A  =  (Nat `  C )   &    |-  H  =  (Homa `  C )   =>    |-  ( F  e.  A  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
 
Theoremhomarw 15996 A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  A  =  (Nat `  C )   &    |-  H  =  (Homa `  C )   =>    |-  ( X H Y )  C_  A
 
Theoremarwdm 15997 The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  A  =  (Nat `  C )   &    |-  B  =  (
 Base `  C )   =>    |-  ( F  e.  A  ->  (domA `  F )  e.  B )
 
Theoremarwcd 15998 The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  A  =  (Nat `  C )   &    |-  B  =  (
 Base `  C )   =>    |-  ( F  e.  A  ->  (coda `  F )  e.  B )
 
Theoremdmaf 15999 The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  A  =  (Nat `  C )   &    |-  B  =  (
 Base `  C )   =>    |-  (domA  |`  A ) : A --> B
 
Theoremcdaf 16000 The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  A  =  (Nat `  C )   &    |-  B  =  (
 Base `  C )   =>    |-  (coda  |`  A ) : A --> B
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-40909
  Copyright terms: Public domain < Previous  Next >