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Theorem List for Metamath Proof Explorer - 15701-15800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisepi2 15701* 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 15702 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 15703 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 15704 Extend class notation with the sections of a morphism.
 class Sect
 
Syntaxcinv 15705 Extend class notation with the inverses of a morphism.
 class Inv
 
Syntaxciso 15706 Extend class notation with the class of all isomorphisms.
 class  Iso
 
Definitiondf-sect 15707* 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. If there there is an arrow  g with  fSect g, the arrow  f is called a section, see definition 7.19 of [Adamek] p. 106. (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 15708* 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. Definition 3.8 of [Adamek] p. 28. (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 15709* Function returning the isomorphisms of the category  c. Definition 3.8 of [Adamek] p. 28, and definition in [Lang] p. 54. (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 15710* 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 15711* 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 15712 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 15713 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 15714 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 15715 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 15716 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 ) )
 
Theoremisofval 15717* Function value of the function returning the isomorphisms of a category. (Contributed by AV, 5-Apr-2017.)
 |-  ( C  e.  Cat  ->  (  Iso  `  C )  =  ( ( x  e. 
 _V  |->  dom  x )  o.  (Inv `  C )
 ) )
 
Theoreminvffval 15718* 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 15719 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 15720 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 15721 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 15722 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 15723 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 15724 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 15725 The isomorphisms are the domain of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 21-May-2020.)
 |-  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 15726 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 15727 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 15728 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 15729 The inverse relation is a bijection from isomorphisms to isomorphisms. This means that every isomorphism  F  e.  ( X I Y ) has a unique inverse, denoted by  ( (Inv `  C
) `  F ). Remark 3.12 of [Adamek] p. 28. (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 15730 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 15731 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 ) ) )
 
Theoremdfiso2 15732* Alternate definition of an isomorphism of a category, according to definition 3.8 in [Adamek] p. 28. (Contributed by AV, 10-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  I  =  (  Iso  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  .1.  =  ( Id `  C )   &    |-  .o.  =  ( <. X ,  Y >. (comp `  C ) X )   &    |-  .*  =  (
 <. Y ,  X >. (comp `  C ) Y )   =>    |-  ( ph  ->  ( F  e.  ( X I Y )  <->  E. g  e.  ( Y H X ) ( ( g  .o.  F )  =  (  .1.  `  X )  /\  ( F  .*  g )  =  (  .1.  `  Y ) ) ) )
 
Theoremdfiso3 15733* Alternate definition of an isomorphism of a category as a section in both directions. (Contributed by AV, 11-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  I  =  (  Iso  `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( F  e.  ( X I Y )  <->  E. g  e.  ( Y H X ) ( g ( Y S X ) F  /\  F ( X S Y ) g ) ) )
 
Theoreminveq 15734 If there are two inverses of an morphism, these inverses are equal. Corollary 3.11 of [Adamek] p. 28. (Contributed by AV, 10-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( ( F ( X N Y ) G 
 /\  F ( X N Y ) K )  ->  G  =  K ) )
 
Theoremisofn 15735 The function value of the function returning the isomorphisms of a category is a function over the square product of the base set of the category. (Contributed by AV, 5-Apr-2017.)
 |-  ( C  e.  Cat  ->  (  Iso  `  C )  Fn  ( ( Base `  C )  X.  ( Base `  C ) ) )
 
Theoremisohom 15736 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 15737 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 15738 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 15739 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 15740 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 15741 An isomorphism in the opposite category. See also remark 3.9 in [Adamek] p. 28. (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 15742 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 15743 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 15744 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 15745 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 "an 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 )
 
Theoremsectid 15746 The identity is a section of itself. (Contributed by AV, 8-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  I  =  ( Id `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( I `  X ) ( X (Sect `  C ) X ) ( I `  X ) )
 
Theoreminvid 15747 The inverse of the identity is the identity. (Contributed by AV, 8-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  I  =  ( Id `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( I `  X ) ( X (Inv `  C ) X ) ( I `  X ) )
 
Theoremidiso 15748 The identity is an isomorphism. Example 3.13 of [Adamek] p. 28. (Contributed by AV, 8-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  I  =  ( Id `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( I `  X )  e.  ( X ( 
 Iso  `  C ) X ) )
 
Theoremidinv 15749 The inverse of the identity is the identity. Example 3.13 of [Adamek] p. 28. (Contributed by AV, 9-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  I  =  ( Id `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( ( X (Inv `  C ) X ) `
  ( I `  X ) )  =  ( I `  X ) )
 
Theoreminvisoinvl 15750 The inverse of an isomorphism  F (which is unique because of invf 15728 and is therefore denoted by  ( ( X N Y ) `  F
), see also remark 3.12 in [Adamek] p. 28) is invers to the isomorphism. (Contributed by AV, 9-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X I Y ) )   =>    |-  ( ph  ->  ( ( X N Y ) `  F ) ( Y N X ) F )
 
Theoreminvisoinvr 15751 The inverse of an isomorphism is invers to the isomorphism. (Contributed by AV, 9-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X I Y ) )   =>    |-  ( ph  ->  F ( X N Y ) ( ( X N Y ) `  F ) )
 
Theoreminvcoisoid 15752 The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 5-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X I Y ) )   &    |-  .1.  =  ( Id `  C )   &    |-  .o.  =  (
 <. X ,  Y >. (comp `  C ) X )   =>    |-  ( ph  ->  ( (
 ( X N Y ) `  F )  .o. 
 F )  =  (  .1.  `  X )
 )
 
Theoremisocoinvid 15753 The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 10-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X I Y ) )   &    |-  .1.  =  ( Id `  C )   &    |-  .o.  =  (
 <. Y ,  X >. (comp `  C ) Y )   =>    |-  ( ph  ->  ( F  .o.  ( ( X N Y ) `  F ) )  =  (  .1.  `  Y ) )
 
Theoremrcaninv 15754 Right cancellation of an inverse of an isomorphism. (Contributed by AV, 5-Apr-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( Y (  Iso  `  C ) X ) )   &    |-  ( ph  ->  G  e.  ( Y ( Hom  `  C ) Z ) )   &    |-  ( ph  ->  H  e.  ( Y ( Hom  `  C ) Z ) )   &    |-  R  =  ( ( Y N X ) `  F )   &    |- 
 .o.  =  ( <. X ,  Y >. (comp `  C ) Z )   =>    |-  ( ph  ->  ( ( G  .o.  R )  =  ( H  .o.  R )  ->  G  =  H ) )
 
8.1.5  Isomorphic objects

In this subsection, the "is isomorphic to" relation between objects of a category  ~=c𝑐 is defined (see df-cic 15756). It is shown that this relation is an equivalence relation, see cicer 15766.

 
Syntaxccic 15755 Extend class notation to include the category isomorphism relation.
 class  ~=c𝑐
 
Definitiondf-cic 15756 Function returning the set of isomorphic objects for each category  c. Definition 3.15 of [Adamek] p. 29. Analogous to the definition of the group isomorphism relation  ~=g𝑔, see df-gic 16979. (Contributed by AV, 4-Apr-2020.)
 |- 
 ~=c𝑐  =  ( c  e.  Cat  |->  ( (  Iso  `  c
 ) supp  (/) ) )
 
Theoremcicfval 15757 The set of isomorphic objects of the category  c. (Contributed by AV, 4-Apr-2020.)
 |-  ( C  e.  Cat  ->  (  ~=c𝑐  `  C )  =  ( (  Iso  `  C ) supp 
 (/) ) )
 
Theorembrcic 15758 The relation "is isomorphic to" for categories. (Contributed by AV, 5-Apr-2020.)
 |-  I  =  (  Iso  `  C )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X (  ~=c𝑐  `  C ) Y  <->  ( X I Y )  =/=  (/) ) )
 
Theoremcic 15759* Objects  X and  Y in a category are isomorphic provided that there is an isomorphism  f : X --> Y, see definition 3.15 of [Adamek] p. 29. (Contributed by AV, 4-Apr-2020.)
 |-  I  =  (  Iso  `  C )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X (  ~=c𝑐  `  C ) Y  <->  E. f  f  e.  ( X I Y ) ) )
 
Theorembrcici 15760 Prove that two objects are isomorphic by an explicit isomorphism. (Contributed by AV, 4-Apr-2020.)
 |-  I  =  (  Iso  `  C )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X I Y ) )   =>    |-  ( ph  ->  X (  ~=c𝑐  `  C ) Y )
 
Theoremcicref 15761 Isomorphism is reflexive. (Contributed by AV, 5-Apr-2020.)
 |-  ( ( C  e.  Cat  /\  O  e.  ( Base `  C ) )  ->  O (  ~=c𝑐  `  C ) O )
 
Theoremciclcl 15762 Isomorphism implies the left side is an object. (Contributed by AV, 5-Apr-2020.)
 |-  ( ( C  e.  Cat  /\  R (  ~=c𝑐  `  C ) S )  ->  R  e.  ( Base `  C )
 )
 
Theoremcicrcl 15763 Isomorphism implies the right side is an object. (Contributed by AV, 5-Apr-2020.)
 |-  ( ( C  e.  Cat  /\  R (  ~=c𝑐  `  C ) S )  ->  S  e.  ( Base `  C )
 )
 
Theoremcicsym 15764 Isomorphism is symmetric. (Contributed by AV, 5-Apr-2020.)
 |-  ( ( C  e.  Cat  /\  R (  ~=c𝑐  `  C ) S )  ->  S (  ~=c𝑐  `  C ) R )
 
Theoremcictr 15765 Isomorphism is transitive. (Contributed by AV, 5-Apr-2020.)
 |-  ( ( C  e.  Cat  /\  R (  ~=c𝑐  `  C ) S  /\  S ( 
 ~=c𝑐  `  C ) T ) 
 ->  R (  ~=c𝑐  `  C ) T )
 
Theoremcicer 15766 Isomorphism is an equivalence relation on objects of a category. Remark 3.16 in [Adamek] p. 29. (Contributed by AV, 5-Apr-2020.)
 |-  ( C  e.  Cat  ->  (  ~=c𝑐  `  C )  Er  ( Base `  C ) )
 
8.1.6  Subcategories
 
Syntaxcssc 15767 Extend class notation to include the subset relation for subcategories.
 class  C_cat
 
Syntaxcresc 15768 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
 class  |`cat
 
Syntaxcsubc 15769 Extend class notation to include the collection of subcategories of a category.
 class Subcat
 
Definitiondf-ssc 15770* 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 15772, 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 15771* 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 15772*  (Subcat `  C
) is the set of all the subcategory specifications of the category  C. Like df-subg 16869, this is not actually a collection of categories (as in definition 4.1(a) of [Adamek] p. 48), but only sets which when given operations from the base category (using df-resc 15771) 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  ( Hom f  `  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 15773 The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |- 
 Rel  C_cat
 
Theorembrssc 15774* 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 15775* An analogue of pwex 4603 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 15776 Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( H  e.  (Subcat `  C )  ->  C  e.  Cat )
 
Theoremsscfn1 15777 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 15778 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 15779 Lemma for ssc1 15781 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  H  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( H  e.  _V  <->  S  e.  _V ) )
 
Theoremisssc 15780* 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 15781 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 15782 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 15783 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 15784 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 15785 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 15786 The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ( A  C_cat  B  /\  B  C_cat  A )  ->  A  =  B )
 
Theoremrescval 15787 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 15788 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 15789 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 15790 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 15791 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 f  `  D ) )
 
Theoremrescco 15792 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 15793 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 15794 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 15795* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  H  =  ( Hom f  `  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 15796* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  H  =  ( Hom f  `  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 ) ) ) ) )
 
Theorem0ssc 15797 For any category  C, the empty set is a subcategory subset of  C. (Contributed by AV, 23-Apr-2020.)
 |-  ( C  e.  Cat  ->  (/)  C_cat 
 ( Hom f  `  C ) )
 
Theorem0subcat 15798 For any category  C, the empty set is a (full) subcategory of  C, see example 4.3(1.a) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.)
 |-  ( C  e.  Cat  ->  (/) 
 e.  (Subcat `  C )
 )
 
Theoremcatsubcat 15799 For any category  C,  C itself is a (full) subcategory of  C, see example 4.3(1.b) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.)
 |-  ( C  e.  Cat  ->  ( Hom f  `  C )  e.  (Subcat `  C ) )
 
Theoremsubcssc 15800 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  =  ( Hom f  `  C )   =>    |-  ( ph  ->  J  C_cat  H )
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