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Theorem List for Metamath Proof Explorer - 13901-14000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxpchomfval 13901* 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 13902 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 13903 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 13904* 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 13905 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 13906 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 13907 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 13908 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 13909 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 13910* 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 13911 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 13912 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 13913* 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 13914 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 13915 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 13916* 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 13917 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 13918 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 13919 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 13920 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 13921* 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 13922 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 13923* 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 13924 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 13925 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 13926 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 13927 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 13928 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 13929 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 13930 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 13931 Extend class notation with the evaluation functor.
 class evalF
 
Syntaxccurf 13932 Extend class notation with the currying of a functor.
 class curryF
 
Syntaxcuncf 13933 Extend class notation with the uncurrying of a functor.
 class uncurryF
 
Syntaxcdiag 13934 Extend class notation to include the diagonal functor.
 class Δfunc
 
Definitiondf-evlf 13935* Define the evaluation functor, which is the extension of the evaluation map  f ,  x  |->  ( f `  x
) of functors, to a functor  ( C --> D )  X.  C --> D. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- evalF  =  ( c  e.  Cat ,  d  e.  Cat  |->  <. ( f  e.  ( c  Func  d ) ,  x  e.  ( Base `  c )  |->  ( ( 1st `  f
 ) `  x )
 ) ,  ( x  e.  ( ( c 
 Func  d )  X.  ( Base `  c ) ) ,  y  e.  (
 ( c  Func  d
 )  X.  ( Base `  c ) )  |->  [_ ( 1st `  x )  /  m ]_ [_ ( 1st `  y )  /  n ]_ ( a  e.  ( m ( c Nat  d ) n ) ,  g  e.  (
 ( 2nd `  x )
 (  Hom  `  c ) ( 2nd `  y
 ) )  |->  ( ( a `  ( 2nd `  y ) ) (
 <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  (
 ( 1st `  m ) `  ( 2nd `  y
 ) ) >. (comp `  d ) ( ( 1st `  n ) `  ( 2nd `  y
 ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y
 ) ) `  g
 ) ) ) )
 >. )
 
Definitiondf-curf 13936* Define the curry functor, which maps a functor  F : C  X.  D --> E to curryF  ( F ) : C --> ( D --> E ). (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- curryF  =  ( e  e.  _V ,  f  e.  _V  |->  [_ ( 1st `  e
 )  /  c ]_ [_ ( 2nd `  e
 )  /  d ]_ <. ( x  e.  ( Base `  c )  |->  <.
 ( y  e.  ( Base `  d )  |->  ( x ( 1st `  f
 ) y ) ) ,  ( y  e.  ( Base `  d ) ,  z  e.  ( Base `  d )  |->  ( g  e.  ( y (  Hom  `  d
 ) z )  |->  ( ( ( Id `  c ) `  x ) ( <. x ,  y >. ( 2nd `  f
 ) <. x ,  z >. ) g ) ) ) >. ) ,  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c )  |->  ( g  e.  ( x (  Hom  `  c
 ) y )  |->  ( z  e.  ( Base `  d )  |->  ( g ( <. x ,  z >. ( 2nd `  f
 ) <. y ,  z >. ) ( ( Id
 `  d ) `  z ) ) ) ) ) >. )
 
Definitiondf-uncf 13937* Define the uncurry functor, which can be defined equationally using evalF. Strictly speaking, the third category argument is not needed, since the resulting functor is extensionally equal regardless, but it is used in the equational definition and is too much work to remove. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |- uncurryF  =  ( c  e.  _V ,  f  e.  _V  |->  ( ( ( c `
  1 ) evalF  ( c `  2 ) )  o.func  (
 ( f  o.func  ( (
 c `  0 )  1stF  ( c `  1
 ) ) ) ⟨,⟩F  ( ( c `  0 )  2ndF  ( c `  1
 ) ) ) ) )
 
Definitiondf-diag 13938* Define the diagonal functor, which is the functor  C --> ( D  Func  C
) whose object part is  x  e.  C  |->  ( y  e.  D  |->  x ). The value of the functor at an object  x is the constant functor which maps all objects in  D to  x and all morphisms to  1 ( x ). The morphism part is a natural transformation between these functors, which takes  f : x --> y to the natural transformation with every component equal to  f. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |- Δfunc  =  ( c  e.  Cat ,  d  e.  Cat  |->  ( <. c ,  d >. curryF  ( c  1stF  d )
 ) )
 
Theoremevlfval 13939* Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  E  =  ( C evalF  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  B  =  ( Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |- 
 .x.  =  (comp `  D )   &    |-  N  =  ( C Nat 
 D )   =>    |-  ( ph  ->  E  =  <. ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f
 ) `  x )
 ) ,  ( x  e.  ( ( C 
 Func  D )  X.  B ) ,  y  e.  ( ( C  Func  D )  X.  B ) 
 |->  [_ ( 1st `  x )  /  m ]_ [_ ( 1st `  y )  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
 ) )  |->  ( ( a `  ( 2nd `  y ) ) (
 <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  (
 ( 1st `  m ) `  ( 2nd `  y
 ) ) >.  .x.  (
 ( 1st `  n ) `  ( 2nd `  y
 ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y
 ) ) `  g
 ) ) ) )
 >. )
 
Theoremevlf2 13940* Value of the evaluation functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  E  =  ( C evalF  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  B  =  ( Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |- 
 .x.  =  (comp `  D )   &    |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  L  =  ( <. F ,  X >. ( 2nd `  E ) <. G ,  Y >. )   =>    |-  ( ph  ->  L  =  ( a  e.  ( F N G ) ,  g  e.  ( X H Y )  |->  ( ( a `  Y ) ( <. ( ( 1st `  F ) `  X ) ,  (
 ( 1st `  F ) `  Y ) >.  .x.  (
 ( 1st `  G ) `  Y ) ) ( ( X ( 2nd `  F ) Y ) `
  g ) ) ) )
 
Theoremevlf2val 13941 Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  E  =  ( C evalF  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  B  =  ( Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |- 
 .x.  =  (comp `  D )   &    |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  L  =  ( <. F ,  X >. ( 2nd `  E ) <. G ,  Y >. )   &    |-  ( ph  ->  A  e.  ( F N G ) )   &    |-  ( ph  ->  K  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( A L K )  =  ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X ) ,  (
 ( 1st `  F ) `  Y ) >.  .x.  (
 ( 1st `  G ) `  Y ) ) ( ( X ( 2nd `  F ) Y ) `
  K ) ) )
 
Theoremevlf1 13942 Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  E  =  ( C evalF  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X ) )
 
Theoremevlfcllem 13943 Lemma for evlfcl 13944. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  E  =  ( C evalF  D )   &    |-  Q  =  ( C FuncCat  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  ( F  e.  ( C 
 Func  D )  /\  X  e.  ( Base `  C )
 ) )   &    |-  ( ph  ->  ( G  e.  ( C 
 Func  D )  /\  Y  e.  ( Base `  C )
 ) )   &    |-  ( ph  ->  ( H  e.  ( C 
 Func  D )  /\  Z  e.  ( Base `  C )
 ) )   &    |-  ( ph  ->  ( A  e.  ( F N G )  /\  K  e.  ( X (  Hom  `  C ) Y ) ) )   &    |-  ( ph  ->  ( B  e.  ( G N H )  /\  L  e.  ( Y (  Hom  `  C ) Z ) ) )   =>    |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  E ) <. H ,  Z >. ) `
  ( <. B ,  L >. ( <. <. F ,  X >. ,  <. G ,  Y >. >. (comp `  ( Q  X.c  C ) ) <. H ,  Z >. ) <. A ,  K >. ) )  =  ( ( (
 <. G ,  Y >. ( 2nd `  E ) <. H ,  Z >. ) `
  <. B ,  L >. ) ( <. ( ( 1st `  E ) `  <. F ,  X >. ) ,  ( ( 1st `  E ) `  <. G ,  Y >. ) >. (comp `  D ) ( ( 1st `  E ) `  <. H ,  Z >. ) ) ( ( <. F ,  X >. ( 2nd `  E ) <. G ,  Y >. ) `  <. A ,  K >. ) ) )
 
Theoremevlfcl 13944 The evaluation functor is a bifunctor (a two-argument functor) with the first parameter taking values in the set of functors  C --> D, and the second parameter in  D. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  E  =  ( C evalF  D )   &    |-  Q  =  ( C FuncCat  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  E  e.  ( ( Q  X.c  C )  Func  D ) )
 
Theoremcurfval 13945* Value of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  G  =  ( <. C ,  D >. curryF  F )   &    |-  A  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )   &    |-  B  =  (
 Base `  D )   &    |-  J  =  (  Hom  `  D )   &    |- 
 .1.  =  ( Id `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  I  =  ( Id `  D )   =>    |-  ( ph  ->  G  =  <. ( x  e.  A  |->  <. ( y  e.  B  |->  ( x ( 1st `  F )
 y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  (
 y J z ) 
 |->  ( (  .1.  `  x ) ( <. x ,  y >. ( 2nd `  F ) <. x ,  z >. ) g ) ) ) >. ) ,  ( x  e.  A ,  y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <. y ,  z >. ) ( I `  z ) ) ) ) ) >. )
 
Theoremcurf1fval 13946* Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  G  =  ( <. C ,  D >. curryF  F )   &    |-  A  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )   &    |-  B  =  (
 Base `  D )   &    |-  J  =  (  Hom  `  D )   &    |- 
 .1.  =  ( Id `  C )   =>    |-  ( ph  ->  ( 1st `  G )  =  ( x  e.  A  |->  <.
 ( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x ) ( <. x ,  y >. ( 2nd `  F ) <. x ,  z >. ) g ) ) ) >. ) )
 
Theoremcurf1 13947* Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  G  =  ( <. C ,  D >. curryF  F )   &    |-  A  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )   &    |-  B  =  (
 Base `  D )   &    |-  ( ph  ->  X  e.  A )   &    |-  K  =  ( ( 1st `  G ) `  X )   &    |-  J  =  ( 
 Hom  `  D )   &    |-  .1.  =  ( Id `  C )   =>    |-  ( ph  ->  K  =  <. ( y  e.  B  |->  ( X ( 1st `  F )
 y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  (
 y J z ) 
 |->  ( (  .1.  `  X ) ( <. X ,  y >. ( 2nd `  F ) <. X ,  z >. ) g ) ) ) >. )
 
Theoremcurf11 13948 Value of the double evaluated curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  G  =  ( <. C ,  D >. curryF  F )   &    |-  A  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )   &    |-  B  =  (
 Base `  D )   &    |-  ( ph  ->  X  e.  A )   &    |-  K  =  ( ( 1st `  G ) `  X )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( ( 1st `  K ) `  Y )  =  ( X ( 1st `  F ) Y ) )
 
Theoremcurf12 13949 The partially evaluated curry functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  G  =  ( <. C ,  D >. curryF  F )   &    |-  A  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )   &    |-  B  =  (
 Base `  D )   &    |-  ( ph  ->  X  e.  A )   &    |-  K  =  ( ( 1st `  G ) `  X )   &    |-  ( ph  ->  Y  e.  B )   &    |-  J  =  (  Hom  `  D )   &    |- 
 .1.  =  ( Id `  C )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  H  e.  ( Y J Z ) )   =>    |-  ( ph  ->  ( ( Y ( 2nd `  K ) Z ) `  H )  =  ( (  .1.  `  X ) (
 <. X ,  Y >. ( 2nd `  F ) <. X ,  Z >. ) H ) )
 
Theoremcurf1cl 13950 The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  G  =  ( <. C ,  D >. curryF  F )   &    |-  A  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )   &    |-  B  =  (
 Base `  D )   &    |-  ( ph  ->  X  e.  A )   &    |-  K  =  ( ( 1st `  G ) `  X )   =>    |-  ( ph  ->  K  e.  ( D  Func  E ) )
 
Theoremcurf2 13951* Value of the curry functor at a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  G  =  ( <. C ,  D >. curryF  F )   &    |-  A  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )   &    |-  B  =  (
 Base `  D )   &    |-  H  =  (  Hom  `  C )   &    |-  I  =  ( Id
 `  D )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  Y  e.  A )   &    |-  ( ph  ->  K  e.  ( X H Y ) )   &    |-  L  =  ( ( X ( 2nd `  G ) Y ) `  K )   =>    |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z >. ( 2nd `  F ) <. Y ,  z >. ) ( I `  z ) ) ) )
 
Theoremcurf2val 13952 Value of a component of the curry functor natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  G  =  ( <. C ,  D >. curryF  F )   &    |-  A  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )   &    |-  B  =  (
 Base `  D )   &    |-  H  =  (  Hom  `  C )   &    |-  I  =  ( Id
 `  D )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  Y  e.  A )   &    |-  ( ph  ->  K  e.  ( X H Y ) )   &    |-  L  =  ( ( X ( 2nd `  G ) Y ) `  K )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( L `  Z )  =  ( K ( <. X ,  Z >. ( 2nd `  F ) <. Y ,  Z >. ) ( I `
  Z ) ) )
 
Theoremcurf2cl 13953 The curry functor at a morphism is a natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  G  =  ( <. C ,  D >. curryF  F )   &    |-  A  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )   &    |-  B  =  (
 Base `  D )   &    |-  H  =  (  Hom  `  C )   &    |-  I  =  ( Id
 `  D )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  Y  e.  A )   &    |-  ( ph  ->  K  e.  ( X H Y ) )   &    |-  L  =  ( ( X ( 2nd `  G ) Y ) `  K )   &    |-  N  =  ( D Nat 
 E )   =>    |-  ( ph  ->  L  e.  ( ( ( 1st `  G ) `  X ) N ( ( 1st `  G ) `  Y ) ) )
 
Theoremcurfcl 13954 The curry functor of a functor  F : C  X.  D
--> E is a functor curryF  ( F ) : C --> ( D --> E ). (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  G  =  ( <. C ,  D >. curryF  F )   &    |-  Q  =  ( D FuncCat  E )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )   =>    |-  ( ph  ->  G  e.  ( C  Func  Q ) )
 
Theoremcurfpropd 13955 If two categories have the same set of objects, morphisms, and compositions, then they curry the same functor to the same result. (Contributed by Mario Carneiro, 26-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.  Cat )   &    |-  ( ph  ->  B  e.  Cat )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  (
 ( A  X.c  C )  Func  E ) )   =>    |-  ( ph  ->  (
 <. A ,  C >. curryF  F )  =  ( <. B ,  D >. curryF  F ) )
 
Theoremuncfval 13956 Value of the uncurry functor, which is the reverse of the curry functor, taking  G : C --> ( D --> E ) to uncurryF  ( G ) : C  X.  D --> E. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  F  =  ( <" C D E "> uncurryF  G )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   &    |-  ( ph  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )   =>    |-  ( ph  ->  F  =  ( ( D evalF  E )  o.func  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
 ) ) )
 
Theoremuncfcl 13957 The uncurry operation takes a functor  F : C --> ( D --> E ) to a functor uncurryF  ( F ) : C  X.  D --> E. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  F  =  ( <" C D E "> uncurryF  G )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   &    |-  ( ph  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )   =>    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )
 
Theoremuncf1 13958 Value of the uncurry functor on an object. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  F  =  ( <" C D E "> uncurryF  G )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   &    |-  ( ph  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )   &    |-  A  =  (
 Base `  C )   &    |-  B  =  ( Base `  D )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X ( 1st `  F ) Y )  =  ( ( 1st `  (
 ( 1st `  G ) `  X ) ) `  Y ) )
 
Theoremuncf2 13959 Value of the uncurry functor on a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  F  =  ( <" C D E "> uncurryF  G )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   &    |-  ( ph  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )   &    |-  A  =  (
 Base `  C )   &    |-  B  =  ( Base `  D )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  Y  e.  B )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  ( ph  ->  Z  e.  A )   &    |-  ( ph  ->  W  e.  B )   &    |-  ( ph  ->  R  e.  ( X H Z ) )   &    |-  ( ph  ->  S  e.  ( Y J W ) )   =>    |-  ( ph  ->  ( R ( <. X ,  Y >. ( 2nd `  F ) <. Z ,  W >. ) S )  =  ( ( ( ( X ( 2nd `  G ) Z ) `  R ) `  W ) (
 <. ( ( 1st `  (
 ( 1st `  G ) `  X ) ) `  Y ) ,  (
 ( 1st `  ( ( 1st `  G ) `  X ) ) `  W ) >. (comp `  E ) ( ( 1st `  ( ( 1st `  G ) `  Z ) ) `  W ) ) ( ( Y ( 2nd `  ( ( 1st `  G ) `  X ) ) W ) `  S ) ) )
 
Theoremcurfuncf 13960 Cancellation of curry with uncurry. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  F  =  ( <" C D E "> uncurryF  G )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   &    |-  ( ph  ->  G  e.  ( C  Func  ( D FuncCat  E ) ) )   =>    |-  ( ph  ->  ( <. C ,  D >. curryF  F )  =  G )
 
Theoremuncfcurf 13961 Cancellation of uncurry with curry. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  G  =  ( <. C ,  D >. curryF  F )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E ) )   =>    |-  ( ph  ->  ( <" C D E "> uncurryF  G )  =  F )
 
Theoremdiagval 13962 Define the diagonal functor, which is the functor  C --> ( D  Func  C
) whose object part is  x  e.  C  |->  ( y  e.  D  |->  x ). We can define this equationally as the currying of the first projection functor, and by expressing it this way we get a quick proof of functoriality. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
 |-  L  =  ( CΔfunc D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
 
Theoremdiagcl 13963 The diagonal functor is a functor from the base category to the functor category. Another way of saying this is that the constant functor  ( y  e.  D  |->  X ) is a construction that is natural in  X (and covariant). (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
 |-  L  =  ( CΔfunc D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  Q  =  ( D FuncCat  C )   =>    |-  ( ph  ->  L  e.  ( C  Func  Q ) )
 
Theoremdiag1cl 13964 The constant functor of  X is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
 |-  L  =  ( CΔfunc D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  A  =  ( Base `  C )   &    |-  ( ph  ->  X  e.  A )   &    |-  K  =  ( ( 1st `  L ) `  X )   =>    |-  ( ph  ->  K  e.  ( D  Func  C ) )
 
Theoremdiag11 13965 Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
 |-  L  =  ( CΔfunc D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  A  =  ( Base `  C )   &    |-  ( ph  ->  X  e.  A )   &    |-  K  =  ( ( 1st `  L ) `  X )   &    |-  B  =  ( Base `  D )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( 1st `  K ) `  Y )  =  X )
 
Theoremdiag12 13966 Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
 |-  L  =  ( CΔfunc D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  A  =  ( Base `  C )   &    |-  ( ph  ->  X  e.  A )   &    |-  K  =  ( ( 1st `  L ) `  X )   &    |-  B  =  ( Base `  D )   &    |-  ( ph  ->  Y  e.  B )   &    |-  J  =  (  Hom  `  D )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( Y J Z ) )   =>    |-  ( ph  ->  ( ( Y ( 2nd `  K ) Z ) `
  F )  =  (  .1.  `  X ) )
 
Theoremdiag2 13967 Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  L  =  ( CΔfunc D )   &    |-  A  =  (
 Base `  C )   &    |-  B  =  ( Base `  D )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  Y  e.  A )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( ( X ( 2nd `  L ) Y ) `
  F )  =  ( B  X.  { F } ) )
 
Theoremdiag2cl 13968 The diagonal functor at a morphism is a natural transformation between constant functors. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  L  =  ( CΔfunc D )   &    |-  A  =  (
 Base `  C )   &    |-  B  =  ( Base `  D )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  Y  e.  A )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  N  =  ( D Nat  C )   =>    |-  ( ph  ->  ( B  X.  { F } )  e.  ( ( ( 1st `  L ) `  X ) N ( ( 1st `  L ) `  Y ) ) )
 
Theoremcurf2ndf 13969 As shown in diagval 13962, the currying of the first projection is the diagonal functor. On the other hand, the currying of the second projection is  x  e.  C  |->  ( y  e.  D  |->  y ), which is a constant functor of the identity functor at  D. (Contributed by Mario Carneiro, 15-Jan-2017.)
 |-  Q  =  ( D FuncCat  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  ( <. C ,  D >. curryF  ( C 
 2ndF  D ) )  =  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )
 
8.4.3  Hom functor
 
Syntaxchof 13970 Extend class notation with the Hom functor.
 class HomF
 
Syntaxcyon 13971 Extend class notation with the Yoneda embedding.
 class Yon
 
Definitiondf-hof 13972* Define the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from  (oppCat `  C )  X.  C to  SetCat, whose object part is the hom-function  Hom, and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- HomF  =  ( c  e.  Cat  |->  <.
 (  Homf  `  c ) ,  [_ ( Base `  c )  /  b ]_ ( x  e.  ( b  X.  b ) ,  y  e.  ( b  X.  b
 )  |->  ( f  e.  ( ( 1st `  y
 ) (  Hom  `  c
 ) ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  c
 ) ( 2nd `  y
 ) )  |->  ( h  e.  ( (  Hom  `  c ) `  x )  |->  ( ( g ( x (comp `  c ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x ) >. (comp `  c ) ( 2nd `  y ) ) f ) ) ) )
 >. )
 
Definitiondf-yon 13973 Define the Yoneda embedding, which is the currying of the (opposite) Hom functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- Yon 
 =  ( c  e. 
 Cat  |->  ( <. c ,  (oppCat `  c ) >. curryF  (HomF `  (oppCat `  c ) ) ) )
 
Theoremhofval 13974* Value of the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from  (oppCat `  C )  X.  C to  SetCat, whose object part is the hom-function 
Hom, and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 15-Jan-2017.)
 |-  M  =  (HomF `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |- 
 .x.  =  (comp `  C )   =>    |-  ( ph  ->  M  =  <. (  Homf  `  C ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B ) 
 |->  ( f  e.  (
 ( 1st `  y ) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
 ) )  |->  ( h  e.  ( H `  x )  |->  ( ( g ( x  .x.  ( 2nd `  y )
 ) h ) (
 <. ( 1st `  y
 ) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) >. )
 
Theoremhof1fval 13975 The object part of the Hom functor is the  Homf operation, which is just a functionalized version of  Hom. That is, it is a two argument function, which maps  X ,  Y to the set of morphisms from  X to  Y. (Contributed by Mario Carneiro, 15-Jan-2017.)
 |-  M  =  (HomF `  C )   &    |-  ( ph  ->  C  e.  Cat )   =>    |-  ( ph  ->  ( 1st `  M )  =  (  Homf  `  C ) )
 
Theoremhof1 13976 The object part of the Hom functor maps  X ,  Y to the set of morphisms from  X to  Y. (Contributed by Mario Carneiro, 15-Jan-2017.)
 |-  M  =  (HomF `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X ( 1st `  M ) Y )  =  ( X H Y ) )
 
Theoremhof2fval 13977* The morphism part of the Hom functor, for morphisms  <. f ,  g >. : <. X ,  Y >. --> <. Z ,  W >. (which since the first argument is contravariant means morphisms  f : Z --> X and  g : Y --> W), yields a function (a morphism of  SetCat) mapping  h : X --> Y to  g  o.  h  o.  f : Z --> W. (Contributed by Mario Carneiro, 15-Jan-2017.)
 |-  M  =  (HomF `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  W  e.  B )   &    |-  .x.  =  (comp `  C )   =>    |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  M ) <. Z ,  W >. )  =  ( f  e.  ( Z H X ) ,  g  e.  ( Y H W ) 
 |->  ( h  e.  ( X H Y )  |->  ( ( g ( <. X ,  Y >.  .x.  W ) h ) ( <. Z ,  X >.  .x.  W ) f ) ) ) )
 
Theoremhof2val 13978* The morphism part of the Hom functor, for morphisms  <. f ,  g >. : <. X ,  Y >. --> <. Z ,  W >. (which since the first argument is contravariant means morphisms  f : Z --> X and  g : Y --> W), yields a function (a morphism of  SetCat) mapping  h : X --> Y to  g  o.  h  o.  f : Z --> W. (Contributed by Mario Carneiro, 15-Jan-2017.)
 |-  M  =  (HomF `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  W  e.  B )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  F  e.  ( Z H X ) )   &    |-  ( ph  ->  G  e.  ( Y H W ) )   =>    |-  ( ph  ->  ( F ( <. X ,  Y >. ( 2nd `  M ) <. Z ,  W >. ) G )  =  ( h  e.  ( X H Y )  |->  ( ( G ( <. X ,  Y >.  .x.  W ) h ) ( <. Z ,  X >.  .x.  W ) F ) ) )
 
Theoremhof2 13979 The morphism part of the Hom functor, for morphisms  <. f ,  g >. : <. X ,  Y >. --> <. Z ,  W >. (which since the first argument is contravariant means morphisms  f : Z --> X and  g : Y --> W), yields a function (a morphism of  SetCat) mapping  h : X --> Y to  g  o.  h  o.  f : Z --> W. (Contributed by Mario Carneiro, 15-Jan-2017.)
 |-  M  =  (HomF `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  W  e.  B )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  F  e.  ( Z H X ) )   &    |-  ( ph  ->  G  e.  ( Y H W ) )   &    |-  ( ph  ->  K  e.  ( X H Y ) )   =>    |-  ( ph  ->  (
 ( F ( <. X ,  Y >. ( 2nd `  M ) <. Z ,  W >. ) G ) `
  K )  =  ( ( G (
 <. X ,  Y >.  .x. 
 W ) K ) ( <. Z ,  X >.  .x.  W ) F ) )
 
Theoremhofcllem 13980 Lemma for hofcl 13981. (Contributed by Mario Carneiro, 15-Jan-2017.)
 |-  M  =  (HomF `  C )   &    |-  O  =  (oppCat `  C )   &    |-  D  =  ( SetCat `  U )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  W  e.  B )   &    |-  ( ph  ->  S  e.  B )   &    |-  ( ph  ->  T  e.  B )   &    |-  ( ph  ->  K  e.  ( Z H X ) )   &    |-  ( ph  ->  L  e.  ( Y H W ) )   &    |-  ( ph  ->  P  e.  ( S H Z ) )   &    |-  ( ph  ->  Q  e.  ( W H T ) )   =>    |-  ( ph  ->  ( ( K ( <. S ,  Z >. (comp `  C ) X ) P ) ( <. X ,  Y >. ( 2nd `  M ) <. S ,  T >. ) ( Q (
 <. Y ,  W >. (comp `  C ) T ) L ) )  =  ( ( P (
 <. Z ,  W >. ( 2nd `  M ) <. S ,  T >. ) Q ) ( <. ( X H Y ) ,  ( Z H W ) >. (comp `  D ) ( S H T ) ) ( K ( <. X ,  Y >. ( 2nd `  M ) <. Z ,  W >. ) L ) ) )
 
Theoremhofcl 13981 Closure of the Hom functor. Note that the codomain is the category  SetCat `  U for any universe  U which contains each Hom-set. This corresponds to the assertion that  C be locally small (with respect to  U). (Contributed by Mario Carneiro, 15-Jan-2017.)
 |-  M  =  (HomF `  C )   &    |-  O  =  (oppCat `  C )   &    |-  D  =  ( SetCat `  U )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   =>    |-  ( ph  ->  M  e.  ( ( O  X.c  C )  Func  D ) )
 
Theoremoppchofcl 13982 Closure of the opposite Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  M  =  (HomF `  O )   &    |-  D  =  ( SetCat `  U )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   =>    |-  ( ph  ->  M  e.  ( ( C  X.c  O )  Func  D ) )
 
Theoremyonval 13983 Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  O  =  (oppCat `  C )   &    |-  M  =  (HomF `  O )   =>    |-  ( ph  ->  Y  =  ( <. C ,  O >. curryF  M ) )
 
Theoremyoncl 13984 The Yoneda embedding is a functor from the category to the category  Q of presheaves on  C. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  Q  =  ( O FuncCat  S )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   =>    |-  ( ph  ->  Y  e.  ( C  Func  Q ) )
 
Theoremyon1cl 13985 The Yoneda embedding at an object of  C is a presheaf on  C, also known as the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  (
 SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   =>    |-  ( ph  ->  (
 ( 1st `  Y ) `  X )  e.  ( O  Func  S ) )
 
Theoremyon11 13986 Value of the Yoneda embedding at an object. The partially evaluated Yoneds embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( 1st `  ( ( 1st `  Y ) `  X ) ) `  Z )  =  ( Z H X ) )
 
Theoremyon12 13987 Value of the Yoneda embedding at a morphism. The partially evaluated Yoneds embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  Z  e.  B )   &    |- 
 .x.  =  (comp `  C )   &    |-  ( ph  ->  W  e.  B )   &    |-  ( ph  ->  F  e.  ( W H Z ) )   &    |-  ( ph  ->  G  e.  ( Z H X ) )   =>    |-  ( ph  ->  ( (
 ( Z ( 2nd `  ( ( 1st `  Y ) `  X ) ) W ) `  F ) `  G )  =  ( G ( <. W ,  Z >.  .x.  X ) F ) )
 
Theoremyon2 13988 Value of the Yoneda embedding at a morphism. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  Z  e.  B )   &    |- 
 .x.  =  (comp `  C )   &    |-  ( ph  ->  W  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Z ) )   &    |-  ( ph  ->  G  e.  ( W H X ) )   =>    |-  ( ph  ->  ( (
 ( ( X ( 2nd `  Y ) Z ) `  F ) `  W ) `  G )  =  ( F ( <. W ,  X >.  .x.  Z ) G ) )
 
Theoremhofpropd 13989 If two categories have the same set of objects, morphisms, and compositions, then they have the same Hom functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  (HomF `  C )  =  (HomF `  D ) )
 
Theoremyonpropd 13990 If two categories have the same set of objects, morphisms, and compositions, then they have the same Yoneda functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  (Yon `  C )  =  (Yon `  D ) )
 
Theoremoppcyon 13991 Value of the opposite Yoneda embedding. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  Y  =  (Yon `  O )   &    |-  M  =  (HomF `  C )   &    |-  ( ph  ->  C  e.  Cat )   =>    |-  ( ph  ->  Y  =  ( <. O ,  C >. curryF  M ) )
 
Theoremoyoncl 13992 The opposite Yoneda embedding is a functor from oppCat `  C to the functor category  C  ->  SetCat. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  Y  =  (Yon `  O )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  S  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  Q  =  ( C FuncCat  S )   =>    |-  ( ph  ->  Y  e.  ( O  Func  Q ) )
 
Theoremoyon1cl 13993 The opposite Yoneda embedding at an object of  C is a functor from  C to Set, also known as the covariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  Y  =  (Yon `  O )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  S  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( ( 1st `  Y ) `  X )  e.  ( C  Func  S ) )
 
Theoremyonedalem1 13994 Lemma for yoneda 14005. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   =>    |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O )  Func  T ) 
 /\  E  e.  (
 ( Q  X.c  O )  Func  T ) ) )
 
Theoremyonedalem21 13995 Lemma for yoneda 14005. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   &    |-  ( ph  ->  F  e.  ( O  Func  S ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( ( 1st `  Y ) `  X ) ( O Nat  S ) F ) )
 
Theoremyonedalem3a 13996* Lemma for yoneda 14005. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   &    |-  ( ph  ->  F  e.  ( O  Func  S ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  M  =  ( f  e.  ( O 
 Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x ) ( O Nat  S ) f )  |->  ( ( a `  x ) `  (  .1.  `  x ) ) ) )   =>    |-  ( ph  ->  ( ( F M X )  =  ( a  e.  (
 ( ( 1st `  Y ) `  X ) ( O Nat  S ) F )  |->  ( ( a `
  X ) `  (  .1.  `  X )
 ) )  /\  ( F M X ) : ( F ( 1st `  Z ) X ) --> ( F ( 1st `  E ) X ) ) )
 
Theoremyonedalem4a 13997* Lemma for yoneda 14005. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   &    |-  ( ph  ->  F  e.  ( O  Func  S ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  N  =  ( f  e.  ( O 
 Func  S ) ,  x  e.  B  |->  ( u  e.  ( ( 1st `  f
 ) `  x )  |->  ( y  e.  B  |->  ( g  e.  (
 y (  Hom  `  C ) x )  |->  ( ( ( x ( 2nd `  f ) y ) `
  g ) `  u ) ) ) ) )   &    |-  ( ph  ->  A  e.  ( ( 1st `  F ) `  X ) )   =>    |-  ( ph  ->  (
 ( F N X ) `  A )  =  ( y  e.  B  |->  ( g  e.  (
 y (  Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
  g ) `  A ) ) ) )
 
Theoremyonedalem4b 13998* Lemma for yoneda 14005. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   &    |-  ( ph  ->  F  e.  ( O  Func  S ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  N  =  ( f  e.  ( O 
 Func  S ) ,  x  e.  B  |->  ( u  e.  ( ( 1st `  f
 ) `  x )  |->  ( y  e.  B  |->  ( g  e.  (
 y (  Hom  `  C ) x )  |->  ( ( ( x ( 2nd `  f ) y ) `
  g ) `  u ) ) ) ) )   &    |-  ( ph  ->  A  e.  ( ( 1st `  F ) `  X ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  G  e.  ( P (  Hom  `  C ) X ) )   =>    |-  ( ph  ->  ( ( ( ( F N X ) `  A ) `  P ) `  G )  =  ( ( ( X ( 2nd `  F ) P ) `  G ) `  A ) )
 
Theoremyonedalem4c 13999* Lemma for yoneda 14005. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   &    |-  ( ph  ->  F  e.  ( O  Func  S ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  N  =  ( f  e.  ( O 
 Func  S ) ,  x  e.  B  |->  ( u  e.  ( ( 1st `  f
 ) `  x )  |->  ( y  e.  B  |->  ( g  e.  (
 y (  Hom  `  C ) x )  |->  ( ( ( x ( 2nd `  f ) y ) `
  g ) `  u ) ) ) ) )   &    |-  ( ph  ->  A  e.  ( ( 1st `  F ) `  X ) )   =>    |-  ( ph  ->  (
 ( F N X ) `  A )  e.  ( ( ( 1st `  Y ) `  X ) ( O Nat  S ) F ) )
 
Theoremyonedalem22 14000 Lemma for yoneda 14005. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   &    |-  ( ph  ->  F  e.  ( O  Func  S ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  G  e.  ( O  Func  S ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  A  e.  ( F ( O Nat  S ) G ) )   &    |-  ( ph  ->  K  e.  ( P (  Hom  `  C ) X ) )   =>    |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  Z ) <. G ,  P >. ) K )  =  ( ( ( P ( 2nd `  Y ) X ) `  K ) ( <. ( ( 1st `  Y ) `  X ) ,  F >. ( 2nd `  H ) <. ( ( 1st `  Y ) `  P ) ,  G >. ) A ) )
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