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Theorem List for Metamath Proof Explorer - 14001-14100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcuncf 14001 Extend class notation with the uncurrying of a functor.
 class uncurryF
 
Syntaxcdiag 14002 Extend class notation to include the diagonal functor.
 class Δfunc
 
Definitiondf-evlf 14003* 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 14004* 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 14005* 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 14006* 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 14007* 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 14008* 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 14009 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 14010 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 14011 Lemma for evlfcl 14012. (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 14012 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 14013* 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 14014* 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 14015* 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 14016 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 14017 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 14018 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 14019* 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 14020 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 14021 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 14022 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 14023 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 14024 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 14025 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 14026 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 14027 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 14028 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 14029 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 14030 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 14031 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 14032 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 14033 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 14034 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 14035 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 14036 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 14037 As shown in diagval 14030, 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 14038 Extend class notation with the Hom functor.
 class HomF
 
Syntaxcyon 14039 Extend class notation with the Yoneda embedding.
 class Yon
 
Definitiondf-hof 14040* 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 14041 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 14042* 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 14043 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 14044 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 14045* 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 14046* 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 14047 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 14048 Lemma for hofcl 14049. (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 14049 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 14050 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 14051 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 14052 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 14053 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 14054 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 14055 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 14056 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 14057 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 14058 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 14059 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 14060 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 14061 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 14062 Lemma for yoneda 14073. (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 14063 Lemma for yoneda 14073. (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 14064* Lemma for yoneda 14073. (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 14065* Lemma for yoneda 14073. (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 14066* Lemma for yoneda 14073. (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 14067* Lemma for yoneda 14073. (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 14068 Lemma for yoneda 14073. (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 ) )
 
Theoremyonedalem3b 14069* Lemma for yoneda 14073. (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 ) )   &    |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x ) ( O Nat  S ) f )  |->  ( ( a `
  x ) `  (  .1.  `  x )
 ) ) )   =>    |-  ( ph  ->  ( ( G M P ) ( <. ( F ( 1st `  Z ) X ) ,  ( G ( 1st `  Z ) P ) >. (comp `  T ) ( G ( 1st `  E ) P ) ) ( A ( <. F ,  X >. ( 2nd `  Z ) <. G ,  P >. ) K ) )  =  ( ( A ( <. F ,  X >. ( 2nd `  E ) <. G ,  P >. ) K ) (
 <. ( F ( 1st `  Z ) X ) ,  ( F ( 1st `  E ) X ) >. (comp `  T ) ( G ( 1st `  E ) P ) ) ( F M X ) ) )
 
Theoremyonedalem3 14070* Lemma for yoneda 14073. (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 )   &    |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x ) ( O Nat  S ) f )  |->  ( ( a `
  x ) `  (  .1.  `  x )
 ) ) )   =>    |-  ( ph  ->  M  e.  ( Z ( ( Q  X.c  O ) Nat  T ) E ) )
 
Theoremyonedainv 14071* The Yoneda Lemma with explicit inverse. (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 )   &    |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x ) ( O Nat  S ) f )  |->  ( ( a `
  x ) `  (  .1.  `  x )
 ) ) )   &    |-  I  =  (Inv `  R )   &    |-  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  ->  M ( Z I E ) N )
 
Theoremyonffthlem 14072* Lemma for yonffth 14074. (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 )   &    |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x ) ( O Nat  S ) f )  |->  ( ( a `
  x ) `  (  .1.  `  x )
 ) ) )   &    |-  I  =  (Inv `  R )   &    |-  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  ->  Y  e.  ( ( C Full  Q )  i^i  ( C Faith  Q ) ) )
 
Theoremyoneda 14073* The Yoneda Lemma. There is a natural isomorphism between the functors  Z and  E, where  Z ( F ,  X ) is the natural transformations from Yon ( X )  =  Hom  (  -  ,  X ) to  F, and  E ( F ,  X )  =  F ( X ) is the evaluation functor. Here we need two universes to state the claim: the smaller universe  U is used for forming the functor category  Q  =  C op  ->  SetCat ( U ), which itself does not (necessarily) live in  U but instead is an element of the larger universe  V. (If  U is a Grothendieck universe, then it will be closed under this "presheaf" operation, and so we can set  U  =  V in this case.) (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 )   &    |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x ) ( O Nat  S ) f )  |->  ( ( a `
  x ) `  (  .1.  `  x )
 ) ) )   &    |-  I  =  (  Iso  `  R )   =>    |-  ( ph  ->  M  e.  ( Z I E ) )
 
Theoremyonffth 14074 The Yoneda Lemma. The Yoneda embedding, the curried Hom functor, is full and faithful, and hence is a representation of the category  C as a full subcategory of the category  Q of presheaves on  C. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  (
 SetCat `  U )   &    |-  Q  =  ( O FuncCat  S )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   =>    |-  ( ph  ->  Y  e.  ( ( C Full  Q )  i^i  ( C Faith  Q ) ) )
 
Theoremyoniso 14075* If the codomain is recoverable from a hom-set, then the Yoneda embedding is injective on objects, and hence is an isomorphism from  C into a full subcategory of a presheaf category. (Contributed by Mario Carneiro, 30-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  (
 SetCat `  U )   &    |-  D  =  (CatCat `  V )   &    |-  B  =  ( Base `  D )   &    |-  I  =  (  Iso  `  D )   &    |-  Q  =  ( O FuncCat  S )   &    |-  E  =  ( Qs 
 ran  ( 1st `  Y ) )   &    |-  ( ph  ->  V  e.  X )   &    |-  ( ph  ->  C  e.  B )   &    |-  ( ph  ->  U  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  E  e.  B )   &    |-  ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) ) 
 ->  ( F `  ( x (  Hom  `  C ) y ) )  =  y )   =>    |-  ( ph  ->  Y  e.  ( C I E ) )
 
PART 9  BASIC ORDER THEORY
 
9.1  Presets and directed sets using extensible structures
 
Syntaxcpreset 14076 Extend class notation with the class of all presets.
 class  Preset
 
Syntaxcdrs 14077 Extend class notation with the class of all directed sets.
 class Dirset
 
Definitiondf-preset 14078* Define the class of preordered sets (presets). A preset is a set equipped with a transitive and reflexive relation.

Preorders are a natural generalization of order for sets where there is a well-defined ordering, but it in some sense "fails to capture the whole story", in that there may be pairs of elements which are indistinguishable under the order. Two elements which are not equal but are less-or-equal to each other behave the same under all order operations and may be thought of as "tied".

A preorder can naturally be strengthened by requiring that there are no ties, resulting in a partial order, or by stating that all comparable pairs of elements are tied, resulting in an equivalence relation. Every preorder naturally factors into these two types; the tied relation on a preorder is an equivalence relation and the quotient under that relation is a partial order. (Contributed by FL, 17-Nov-2014.) (Revised by Stefan O'Rear, 31-Jan-2015.)

 |-  Preset  =  { f  | 
 [. ( Base `  f
 )  /  b ]. [. ( le `  f
 )  /  r ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r z )  ->  x r z ) ) }
 
Definitiondf-drs 14079* Define the class of directed sets. A directed set is a nonempty preordered set where every pair of elements have some upper bound. Note that it is not required that there exist a least upper bound.

There is no consensus in the literature over whether directed sets are allowed to be empty. It is slightly more convenient for us if they are not. (Contributed by Stefan O'Rear, 1-Feb-2015.)

 |- Dirset  =  { f  e.  Preset  | 
 [. ( Base `  f
 )  /  b ]. [. ( le `  f
 )  /  r ]. ( b  =/=  (/)  /\  A. x  e.  b  A. y  e.  b  E. z  e.  b  ( x r z  /\  y r z ) ) }
 
Theoremisprs 14080* Property of being a preordered set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( K  e.  Preset  <->  ( K  e.  _V  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .<_  x 
 /\  ( ( x 
 .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) ) ) )
 
Theoremprslem 14081 Lemma for prsref 14082 and prstr 14083. (Contributed by Mario Carneiro, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )
 
Theoremprsref 14082 Less-or-equal is reflexive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Preset  /\  X  e.  B ) 
 ->  X  .<_  X )
 
Theoremprstr 14083 Less-or-equal is transitive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  .<_  Y  /\  Y  .<_  Z ) ) 
 ->  X  .<_  Z )
 
Theoremisdrs 14084* Property of being a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( K  e. Dirset  <->  ( K  e.  Preset  /\  B  =/=  (/)  /\  A. x  e.  B  A. y  e.  B  E. z  e.  B  ( x  .<_  z 
 /\  y  .<_  z ) ) )
 
Theoremdrsdir 14085* Direction of a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e. Dirset  /\  X  e.  B  /\  Y  e.  B )  ->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) )
 
Theoremdrsprs 14086 A directed set is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( K  e. Dirset  ->  K  e.  Preset  )
 
Theoremdrsbn0 14087 The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   =>    |-  ( K  e. Dirset  ->  B  =/=  (/) )
 
Theoremdrsdirfi 14088* Any finite number of elements in a directed set have a common upper bound. Here is where the non-emptiness constraint in df-drs 14079 first comes into play; without it we would need an additional constraint that  X not be empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e. Dirset  /\  X  C_  B  /\  X  e.  Fin )  ->  E. y  e.  B  A. z  e.  X  z 
 .<_  y )
 
Theoremisdrs2 14089* Directed sets may be defined in terms of finite subsets. Again, without nonemptiness we would need to restrict to nonempty subsets here. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( K  e. Dirset  <->  ( K  e.  Preset  /\ 
 A. x  e.  ( ~P B  i^i  Fin ) E. y  e.  B  A. z  e.  x  z 
 .<_  y ) )
 
9.2  Posets and lattices using extensible structures
 
9.2.1  Posets
 
Syntaxcpo 14090 Extend class notation with the class of posets.
 class  Poset
 
Syntaxcplt 14091 Extend class notation with less-than for posets.
 class  lt
 
Syntaxclub 14092 Extend class notation with poset least upper bound.
 class  lub
 
Syntaxcglb 14093 Extend class notation with poset greatest lower bound.
 class  glb
 
Syntaxcjn 14094 Extend class notation with poset join.
 class  join
 
Syntaxcmee 14095 Extend class notation with poset meet.
 class  meet
 
Definitiondf-poset 14096* Define the class of posets. Definition of poset in [Crawley] p. 1. Note that Crawley-Dilworth require that a poset base set be nonempty, but we follow the convention of most authors who don't make this a requirement.

The quantifiers  E. b E. r provide a notational shorthand to allow us to refer to the base and ordering relation as  b and  r the definition rather than having to repeat  (
Base `  f ) and  ( le `  f ) throughout. These quantifiers can be eliminated with ceqsex2v 2838 and related theorems. (Contributed by NM, 18-Oct-2012.)

 |- 
 Poset  =  { f  |  E. b E. r
 ( b  =  (
 Base `  f )  /\  r  =  ( le `  f )  /\  A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r x )  ->  x  =  y )  /\  (
 ( x r y 
 /\  y r z )  ->  x r
 z ) ) ) }
 
Theoremispos 14097* The predicate "is a poset." (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 4-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( K  e.  Poset  <->  ( K  e.  _V  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .<_  x 
 /\  ( ( x 
 .<_  y  /\  y  .<_  x )  ->  x  =  y )  /\  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) ) ) )
 
Theoremispos2 14098* A poset is an antisymmetric preset.

EDITORIAL: could become the definition of poset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( K  e.  Poset  <->  ( K  e.  Preset  /\  A. x  e.  B  A. y  e.  B  ( ( x 
 .<_  y  /\  y  .<_  x )  ->  x  =  y ) ) )
 
Theoremposprs 14099 A poset is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( K  e.  Poset  ->  K  e.  Preset  )
 
Theoremposi 14100 Lemma for poset properties. (Contributed by NM, 11-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )  /\  (
 ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )
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