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Theorem List for Metamath Proof Explorer - 13601-13700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmonhom 13601 A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  M  =  (Mono `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X M Y ) 
 C_  ( X H Y ) )
 
Theoremmoni 13602 Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  M  =  (Mono `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X M Y ) )   &    |-  ( ph  ->  G  e.  ( Z H X ) )   &    |-  ( ph  ->  K  e.  ( Z H X ) )   =>    |-  ( ph  ->  ( ( F ( <. Z ,  X >.  .x.  Y ) G )  =  ( F ( <. Z ,  X >.  .x.  Y ) K )  <->  G  =  K ) )
 
Theoremmonpropd 13603 If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  (Mono `  C )  =  (Mono `  D ) )
 
Theoremoppcmon 13604 A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  M  =  (Mono `  O )   &    |-  E  =  (Epi `  C )   =>    |-  ( ph  ->  ( X M Y )  =  ( Y E X ) )
 
Theoremoppcepi 13605 An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  E  =  (Epi `  O )   &    |-  M  =  (Mono `  C )   =>    |-  ( ph  ->  ( X E Y )  =  ( Y M X ) )
 
Theoremisepi 13606* Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  E  =  (Epi `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( F  e.  ( X E Y )  <->  ( F  e.  ( X H Y ) 
 /\  A. z  e.  B  Fun  `' ( g  e.  ( Y H z )  |->  ( g ( <. X ,  Y >.  .x.  z ) F ) ) ) ) )
 
Theoremisepi2 13607* Write out the epimorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  E  =  (Epi `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( F  e.  ( X E Y )  <->  ( F  e.  ( X H Y ) 
 /\  A. z  e.  B  A. g  e.  ( Y H z ) A. h  e.  ( Y H z ) ( ( g ( <. X ,  Y >.  .x.  z
 ) F )  =  ( h ( <. X ,  Y >.  .x.  z
 ) F )  ->  g  =  h )
 ) ) )
 
Theoremepihom 13608 An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  E  =  (Epi `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X E Y ) 
 C_  ( X H Y ) )
 
Theoremepii 13609 Property of an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  E  =  (Epi `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X E Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   &    |-  ( ph  ->  K  e.  ( Y H Z ) )   =>    |-  ( ph  ->  ( ( G ( <. X ,  Y >.  .x.  Z ) F )  =  ( K ( <. X ,  Y >.  .x.  Z ) F )  <->  G  =  K ) )
 
8.1.4  Sections, inverses, isomorphisms
 
Syntaxcsect 13610 Extend class notation with the sections of a morphism.
 class Sect
 
Syntaxcinv 13611 Extend class notation with the inverses of a morphism.
 class Inv
 
Syntaxciso 13612 Extend class notation with the class of all isomorphisms.
 class  Iso
 
Definitiondf-sect 13613* Function returning the section relation in a category. Given arrows  f : X --> Y and  g : Y --> X, we say  fSect g, that is,  f is a section of  g, if  g  o.  f  =  1 `  X. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- Sect  =  ( c  e.  Cat  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c )  |->  { <. f ,  g >.  |  [. (  Hom  `  c )  /  h ]. ( ( f  e.  ( x h y )  /\  g  e.  ( y h x ) )  /\  ( g ( <. x ,  y >. (comp `  c ) x ) f )  =  ( ( Id `  c
 ) `  x )
 ) } ) )
 
Definitiondf-inv 13614* The inverse relation in a category. Given arrows  f : X --> Y and  g : Y --> X, we say  gInv f, that is,  g is an inverse of  f, if  g is a section of  f and  f is a section of  g. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
 |- Inv 
 =  ( c  e. 
 Cat  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c )  |->  ( ( x (Sect `  c ) y )  i^i  `' ( y (Sect `  c ) x ) ) ) )
 
Definitiondf-iso 13615* Function returning the isomorphisms of the category  c. The Joy of Cats p. 28. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 2-Jan-2017.)
 |- 
 Iso  =  ( c  e.  Cat  |->  ( ( x  e.  _V  |->  dom  x )  o.  (Inv `  c
 ) ) )
 
Theoremsectffval 13616* Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  S  =  ( x  e.  B ,  y  e.  B  |->  {
 <. f ,  g >.  |  ( ( f  e.  ( x H y )  /\  g  e.  ( y H x ) )  /\  (
 g ( <. x ,  y >.  .x.  x )
 f )  =  (  .1.  `  x )
 ) } ) )
 
Theoremsectfval 13617* Value of the section relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X S Y )  =  { <. f ,  g >.  |  ( ( f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) }
 )
 
Theoremsectss 13618 The section relation is a relation between morphisms from  X to  Y and morphisms from  Y to  X. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X S Y )  C_  ( ( X H Y )  X.  ( Y H X ) ) )
 
Theoremissect 13619 The property " F is a section of  G". (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( F ( X S Y ) G  <->  ( F  e.  ( X H Y ) 
 /\  G  e.  ( Y H X )  /\  ( G ( <. X ,  Y >.  .x.  X ) F )  =  (  .1.  `  X ) ) ) )
 
Theoremissect2 13620 Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H X ) )   =>    |-  ( ph  ->  ( F ( X S Y ) G  <->  ( G (
 <. X ,  Y >.  .x. 
 X ) F )  =  (  .1.  `  X ) ) )
 
Theoremsectcan 13621 If  G is a section of  F and  F is a section of  H, then  G  =  H. Proposition 3.10 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  G ( X S Y ) F )   &    |-  ( ph  ->  F ( Y S X ) H )   =>    |-  ( ph  ->  G  =  H )
 
Theoremsectco 13622 Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  .x.  =  (comp `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F ( X S Y ) G )   &    |-  ( ph  ->  H ( Y S Z ) K )   =>    |-  ( ph  ->  ( H ( <. X ,  Y >.  .x.  Z ) F ) ( X S Z ) ( G ( <. Z ,  Y >.  .x.  X ) K ) )
 
Theoreminvffval 13623* Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  S  =  (Sect `  C )   =>    |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
 
Theoreminvfval 13624 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  S  =  (Sect `  C )   =>    |-  ( ph  ->  ( X N Y )  =  ( ( X S Y )  i^i  `' ( Y S X ) ) )
 
Theoremisinv 13625 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  S  =  (Sect `  C )   =>    |-  ( ph  ->  ( F ( X N Y ) G  <->  ( F ( X S Y ) G  /\  G ( Y S X ) F ) ) )
 
Theoreminvss 13626 The inverse relation is a relation between morphisms  F : X --> Y and their inverses  G : Y --> X. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  H  =  ( 
 Hom  `  C )   =>    |-  ( ph  ->  ( X N Y )  C_  ( ( X H Y )  X.  ( Y H X ) ) )
 
Theoreminvsym 13627 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( F ( X N Y ) G  <->  G ( Y N X ) F ) )
 
Theoreminvsym2 13628 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
 
Theoreminvfun 13629 The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  Fun  ( X N Y ) )
 
Theoremisoval 13630 The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   =>    |-  ( ph  ->  ( X I Y )  =  dom  (  X N Y ) )
 
Theoreminviso1 13631 If  G is an inverse to  F, then  F is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  ( ph  ->  F ( X N Y ) G )   =>    |-  ( ph  ->  F  e.  ( X I Y ) )
 
Theoreminviso2 13632 If  G is an inverse to  F, then  G is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  ( ph  ->  F ( X N Y ) G )   =>    |-  ( ph  ->  G  e.  ( Y I X ) )
 
Theoreminvf 13633 The inverse relation is a function from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   =>    |-  ( ph  ->  ( X N Y ) : ( X I Y ) --> ( Y I X ) )
 
Theoreminvf1o 13634 The inverse relation is a bijection from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   =>    |-  ( ph  ->  ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )
 
Theoreminvinv 13635 The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  ( ph  ->  F  e.  ( X I Y ) )   =>    |-  ( ph  ->  ( ( Y N X ) `  ( ( X N Y ) `  F ) )  =  F )
 
Theoreminvco 13636 The composition of two isomorphisms is an isomorphism, and the inverse is the composition of the individual inverses. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  ( ph  ->  F  e.  ( X I Y ) )   &    |-  .x. 
 =  (comp `  C )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  G  e.  ( Y I Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F ) ( X N Z ) ( ( ( X N Y ) `  F ) ( <. Z ,  Y >.  .x.  X )
 ( ( Y N Z ) `  G ) ) )
 
Theoremisohom 13637 An isomorphism is a homomorphism. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  I  =  (  Iso  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X I Y )  C_  ( X H Y ) )
 
Theoremisoco 13638 The composition of two isomorphisms is an isomorphism. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  .x.  =  (comp `  C )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X I Y ) )   &    |-  ( ph  ->  G  e.  ( Y I Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X I Z ) )
 
Theoremoppcsect 13639 A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  O  =  (oppCat `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  S  =  (Sect `  C )   &    |-  T  =  (Sect `  O )   =>    |-  ( ph  ->  ( F ( X T Y ) G  <->  G ( X S Y ) F ) )
 
Theoremoppcsect2 13640 A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  O  =  (oppCat `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  S  =  (Sect `  C )   &    |-  T  =  (Sect `  O )   =>    |-  ( ph  ->  ( X T Y )  =  `' ( X S Y ) )
 
Theoremoppcinv 13641 An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  O  =  (oppCat `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  (Inv `  C )   &    |-  J  =  (Inv `  O )   =>    |-  ( ph  ->  ( X J Y )  =  ( Y I X ) )
 
Theoremoppciso 13642 An isomorphism in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  O  =  (oppCat `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  J  =  (  Iso  `  O )   =>    |-  ( ph  ->  ( X J Y )  =  ( Y I X ) )
 
Theoremsectmon 13643 If  F is a section of  G, then  F is a monomorphism. A monomorphism that arises from a section is also known as a split monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  M  =  (Mono `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F ( X S Y ) G )   =>    |-  ( ph  ->  F  e.  ( X M Y ) )
 
Theoremmonsect 13644 If  F is a monomorphism and  G is a section of  F, then  G is an inverse of  F and they are both isomorphisms. This is also stated as "a monomorphism which is also a split epimorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  M  =  (Mono `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  F  e.  ( X M Y ) )   &    |-  ( ph  ->  G ( Y S X ) F )   =>    |-  ( ph  ->  F ( X N Y ) G )
 
Theoremsectepi 13645 If  F is a section of  G, then  G is an epimorphism. An epimorphism that arises from a section is also known as a split epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  E  =  (Epi `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F ( X S Y ) G )   =>    |-  ( ph  ->  G  e.  ( Y E X ) )
 
Theoremepisect 13646 If  F is an epimorphism and  F is a section of  G, then  G is an inverse of  F and they are both isomorphisms. This is also stated as "a epimorphism which is also a split monomorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  E  =  (Epi `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  F  e.  ( X E Y ) )   &    |-  ( ph  ->  F ( X S Y ) G )   =>    |-  ( ph  ->  F ( X N Y ) G )
 
8.1.5  Subcategories
 
Syntaxcssc 13647 Extend class notation to include the subset relation for subcategories.
 class  C_cat
 
Syntaxcresc 13648 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
 class  |`cat
 
Syntaxcsubc 13649 Extend class notation to include the collection of subcategories of a category.
 class Subcat
 
Definitiondf-ssc 13650* Define the subset relation for subcategories. Despite the name, this is not really a "category-aware" definition, which is to say it makes no explicit references to homsets or composition; instead this is a subset-like relation on the functions that are used as subcategory specifications in df-subc 13652, which makes it play an analogous role to the subset relation applied to the subgroups of a group. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  C_cat 
 =  { <. h ,  j >.  |  E. t
 ( j  Fn  (
 t  X.  t )  /\  E. s  e.  ~P  t h  e.  X_ x  e.  ( s  X.  s
 ) ~P ( j `
  x ) ) }
 
Definitiondf-resc 13651* Define the restriction of a category to a given set of arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  |`cat 
 =  ( c  e. 
 _V ,  h  e. 
 _V  |->  ( ( cs  dom 
 dom  h ) sSet  <. ( 
 Hom  `  ndx ) ,  h >. ) )
 
Definitiondf-subc 13652*  (Subcat `  C
) is the set of all the subcategory specifications of the category  C. Like df-subg 14581, this is not actually a collection of categories, but only sets which when given operations from the base category (using df-resc 13651) form a category. All the objects and all the morphisms of the subcategory belong to the supercategory. The identity of an object, the domain and the codomain of a morphism are the same in the subcategory and the supercategory. The composition of the subcategory is a restriction of the composition of the supercategory. (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 4-Jan-2017.)
 |- Subcat  =  ( c  e.  Cat  |->  { h  |  ( h 
 C_cat  (  Homf  `  c )  /\  [. dom  dom 
 h  /  s ]. A. x  e.  s  ( ( ( Id `  c ) `  x )  e.  ( x h x )  /\  A. y  e.  s  A. z  e.  s  A. f  e.  ( x h y ) A. g  e.  ( y h z ) ( g ( <. x ,  y >. (comp `  c
 ) z ) f )  e.  ( x h z ) ) ) } )
 
Theoremsscrel 13653 The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |- 
 Rel  C_cat
 
Theorembrssc 13654* The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( H  C_cat  J  <->  E. t ( J  Fn  ( t  X.  t )  /\  E. s  e.  ~P  t H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x ) ) )
 
Theoremsscpwex 13655* An analogue of pwex 4165 for the subcategory subset relation: The collection of subcategory subsets of a given set  J is a set. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |- 
 { h  |  h  C_cat  J }  e.  _V
 
Theoremsubcrcl 13656 Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( H  e.  (Subcat `  C )  ->  C  e.  Cat )
 
Theoremsscfn1 13657 The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  H  C_cat  J )   &    |-  ( ph  ->  S  =  dom  dom  H )   =>    |-  ( ph  ->  H  Fn  ( S  X.  S ) )
 
Theoremsscfn2 13658 The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  H  C_cat  J )   &    |-  ( ph  ->  T  =  dom  dom  J )   =>    |-  ( ph  ->  J  Fn  ( T  X.  T ) )
 
Theoremssclem 13659 Lemma for ssc1 13661 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  H  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( H  e.  _V  <->  S  e.  _V ) )
 
Theoremisssc 13660* Value of the subcategory subset relation when the arguments are known functions. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  J  Fn  ( T  X.  T ) )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  ( H  C_cat  J  <->  ( S  C_  T  /\  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) ) ) )
 
Theoremssc1 13661 Infer subset relation on objects from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  J  Fn  ( T  X.  T ) )   &    |-  ( ph  ->  H  C_cat  J )   =>    |-  ( ph  ->  S  C_  T )
 
Theoremssc2 13662 Infer subset relation on morphisms from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  H 
 C_cat  J )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  ( X H Y )  C_  ( X J Y ) )
 
Theoremsscres 13663 Any function restricted to a square domain is a subcategory subset of the original. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ( H  Fn  ( S  X.  S ) 
 /\  S  e.  V )  ->  ( H  |`  ( T  X.  T ) ) 
 C_cat  H )
 
Theoremsscid 13664 The subcategory subset relation is reflexive. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ( H  Fn  ( S  X.  S ) 
 /\  S  e.  V )  ->  H  C_cat  H )
 
Theoremssctr 13665 The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ( A  C_cat  B  /\  B  C_cat  C )  ->  A  C_cat  C )
 
Theoremssceq 13666 The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ( A  C_cat  B  /\  B  C_cat  A )  ->  A  =  B )
 
Theoremrescval 13667 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   =>    |-  ( ( C  e.  V  /\  H  e.  W )  ->  D  =  ( ( Cs  dom  dom  H ) sSet  <.
 (  Hom  `  ndx ) ,  H >. ) )
 
Theoremrescval2 13668 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  D  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
 
Theoremrescbas 13669 Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   =>    |-  ( ph  ->  S  =  ( Base `  D ) )
 
Theoremreschom 13670 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   =>    |-  ( ph  ->  H  =  (  Hom  `  D ) )
 
Theoremreschomf 13671 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   =>    |-  ( ph  ->  H  =  (  Homf  `  D ) )
 
Theoremrescco 13672 Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   &    |-  .x.  =  (comp `  C )   =>    |-  ( ph  ->  .x. 
 =  (comp `  D ) )
 
Theoremrescabs 13673 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  J  Fn  ( T  X.  T ) )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  T 
 C_  S )   =>    |-  ( ph  ->  ( ( C  |`cat  H )  |`cat  J )  =  ( C  |`cat  J ) )
 
Theoremrescabs2 13674 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  J  Fn  ( T  X.  T ) )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  T  C_  S )   =>    |-  ( ph  ->  (
 ( Cs  S )  |`cat  J )  =  ( C  |`cat  J )
 )
 
Theoremissubc 13675* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  H  =  (  Homf  `  C )   &    |- 
 .1.  =  ( Id `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  S  =  dom  dom 
 J )   =>    |-  ( ph  ->  ( J  e.  (Subcat `  C ) 
 <->  ( J  C_cat  H  /\  A. x  e.  S  ( (  .1.  `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
 <. x ,  y >.  .x.  z ) f )  e.  ( x J z ) ) ) ) )
 
Theoremissubc2 13676* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  H  =  (  Homf  `  C )   &    |- 
 .1.  =  ( Id `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  H  /\  A. x  e.  S  ( (  .1.  `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
 <. x ,  y >.  .x.  z ) f )  e.  ( x J z ) ) ) ) )
 
Theoremsubcssc 13677 An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  H  =  (  Homf  `  C )   =>    |-  ( ph  ->  J  C_cat  H )
 
Theoremsubcfn 13678 An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  S  =  dom  dom  J )   =>    |-  ( ph  ->  J  Fn  ( S  X.  S ) )
 
Theoremsubcss1 13679 The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  B  =  (
 Base `  C )   =>    |-  ( ph  ->  S 
 C_  B )
 
Theoremsubcss2 13680 The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  ( X J Y )  C_  ( X H Y ) )
 
Theoremsubcidcl 13681 The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  .1.  =  ( Id `  C )   =>    |-  ( ph  ->  (  .1.  `  X )  e.  ( X J X ) )
 
Theoremsubccocl 13682 A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  Y  e.  S )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ph  ->  F  e.  ( X J Y ) )   &    |-  ( ph  ->  G  e.  ( Y J Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) )
 
Theoremsubccatid 13683* A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  .1.  =  ( Id `  C )   =>    |-  ( ph  ->  ( D  e.  Cat  /\  ( Id `  D )  =  ( x  e.  S  |->  (  .1.  `  x )
 ) ) )
 
Theoremsubcid 13684 The identity in a subcategory is the same as the original category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  X  e.  S )   =>    |-  ( ph  ->  (  .1.  `  X )  =  ( ( Id `  D ) `  X ) )
 
Theoremsubccat 13685 A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   =>    |-  ( ph  ->  D  e.  Cat )
 
Theoremissubc3 13686* Alternate definition of a subcategory, as a subset of the category which is itself a category. The assumption that the identity be closed is necessary just as in the case of a monoid, issubm2 14389, for the same reasons, since categories are a generalization of monoids. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  H  =  (  Homf  `  C )   &    |- 
 .1.  =  ( Id `  C )   &    |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  H  /\  A. x  e.  S  (  .1.  `  x )  e.  ( x J x )  /\  D  e.  Cat ) ) )
 
Theoremfullsubc 13687 The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory). (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Homf  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  S  C_  B )   =>    |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  e.  (Subcat `  C )
 )
 
Theoremfullresc 13688 The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Homf  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  S  C_  B )   &    |-  D  =  ( Cs  S )   &    |-  E  =  ( C  |`cat  ( H  |`  ( S  X.  S ) ) )   =>    |-  ( ph  ->  (
 (  Homf  `  D )  =  ( 
 Homf  `  E )  /\  (compf `  D )  =  (compf `  E ) ) )
 
Theoremresscat 13689 A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ( C  e.  Cat  /\  S  e.  V ) 
 ->  ( Cs  S )  e.  Cat )
 
Theoremsubsubc 13690 A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   =>    |-  ( H  e.  (Subcat `  C )  ->  ( J  e.  (Subcat `  D ) 
 <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H ) ) )
 
8.1.6  Functors
 
Syntaxcfunc 13691 Extend class notation with the class of all functors.
 class  Func
 
Syntaxcidfu 13692 Extend class notation with identity functor.
 class idfunc
 
Syntaxccofu 13693 Extend class notation with functor composition.
 class  o.func
 
Syntaxcresf 13694 Extend class notation to include restriction of a functor to a subcategory.
 class  |`f
 
Definitiondf-func 13695* Function returning all the functors from a category  t to a category  u. Intuitively a functor associates any morphism of  t to a morphism of  u, any object of  t to an object of  u, and respects the identity, the composition, the domain and the codomain. Here to capture the idea that a functor associates any object of  t to an object of  u we write it associates any identity of  t to an identity of  u which simplifies the definition. (Contributed by FL, 10-Feb-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
 |- 
 Func  =  ( t  e.  Cat ,  u  e. 
 Cat  |->  { <. f ,  g >.  |  [. ( Base `  t )  /  b ]. ( f : b --> ( Base `  u )  /\  g  e.  X_ z  e.  ( b  X.  b
 ) ( ( ( f `  ( 1st `  z ) ) ( 
 Hom  `  u )
 ( f `  ( 2nd `  z ) ) )  ^m  ( ( 
 Hom  `  t ) `  z ) )  /\  A. x  e.  b  ( ( ( x g x ) `  (
 ( Id `  t
 ) `  x )
 )  =  ( ( Id `  u ) `
  ( f `  x ) )  /\  A. y  e.  b  A. z  e.  b  A. m  e.  ( x (  Hom  `  t )
 y ) A. n  e.  ( y (  Hom  `  t ) z ) ( ( x g z ) `  ( n ( <. x ,  y >. (comp `  t
 ) z ) m ) )  =  ( ( ( y g z ) `  n ) ( <. ( f `
  x ) ,  ( f `  y
 ) >. (comp `  u ) ( f `  z ) ) ( ( x g y ) `  m ) ) ) ) }
 )
 
Definitiondf-idfu 13696* Define the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- idfunc  =  ( t  e.  Cat  |->  [_ ( Base `  t )  /  b ]_ <. (  _I  |`  b ) ,  (
 z  e.  ( b  X.  b )  |->  (  _I  |`  ( (  Hom  `  t ) `  z ) ) )
 >. )
 
Definitiondf-cofu 13697* Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- 
 o.func  =  ( g  e.  _V ,  f  e.  _V  |->  <.
 ( ( 1st `  g
 )  o.  ( 1st `  f ) ) ,  ( x  e.  dom  dom  ( 2nd `  f
 ) ,  y  e. 
 dom  dom  ( 2nd `  f
 )  |->  ( ( ( ( 1st `  f
 ) `  x )
 ( 2nd `  g )
 ( ( 1st `  f
 ) `  y )
 )  o.  ( x ( 2nd `  f
 ) y ) ) ) >. )
 
Definitiondf-resf 13698* Define the restriction of a functor to a subcategory (analogue of df-res 4681). (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  |`f  =  ( f  e.  _V ,  h  e.  _V  |->  <.
 ( ( 1st `  f
 )  |`  dom  dom  h ) ,  ( x  e. 
 dom  h  |->  ( ( ( 2nd `  f
 ) `  x )  |`  ( h `  x ) ) ) >. )
 
Theoremrelfunc 13699 The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 Rel  ( D  Func  E )
 
Theoremfuncrcl 13700 Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( F  e.  ( D  Func  E )  ->  ( D  e.  Cat  /\  E  e.  Cat )
 )
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