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Theorem List for Metamath Proof Explorer - 13501-13600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
7.2.3  Algebraic closure systems
 
Theoremisacs 13501* A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  E. f ( f : ~P X --> ~P X  /\  A. s  e.  ~P  X ( s  e.  C  <->  U. ( f "
 ( ~P s  i^i 
 Fin ) )  C_  s ) ) ) )
 
Theoremacsmre 13502 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  ->  C  e.  (Moore `  X )
 )
 
Theoremisacs2 13503* In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( s  e.  C  <->  A. y  e.  ( ~P s  i^i  Fin )
 ( F `  y
 )  C_  s )
 ) )
 
Theoremacsfiel 13504* A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (ACS `  X )  ->  ( S  e.  C  <->  ( S  C_  X  /\  A. y  e.  ( ~P S  i^i  Fin ) ( F `  y )  C_  S ) ) )
 
Theoremacsfiel2 13505* A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  ( S  e.  C  <->  A. y  e.  ( ~P S  i^i  Fin )
 ( F `  y
 )  C_  S )
 )
 
Theoremacsmred 13506 An algebraic closure system is also a Moore system. Deduction form of acsmre 13502. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   =>    |-  ( ph  ->  A  e.  (Moore `  X )
 )
 
Theoremisacs1i 13507* A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( X  e.  V  /\  F : ~P X
 --> ~P X )  ->  { s  e.  ~P X  |  U. ( F
 " ( ~P s  i^i  Fin ) )  C_  s }  e.  (ACS `  X ) )
 
Theoremmreacs 13508 Algebraicity is a composible property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  e.  V  ->  (ACS `  X )  e.  (Moore `  ~P X ) )
 
Theoremacsfn 13509* Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( ( X  e.  V  /\  K  e.  X )  /\  ( T  C_  X  /\  T  e.  Fin ) )  ->  { a  e.  ~P X  |  ( T  C_  a  ->  K  e.  a ) }  e.  (ACS `  X ) )
 
Theoremacsfn0 13510* Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( X  e.  V  /\  K  e.  X )  ->  { a  e. 
 ~P X  |  K  e.  a }  e.  (ACS `  X ) )
 
Theoremacsfn1 13511* Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( X  e.  V  /\  A. b  e.  X  E  e.  X )  ->  { a  e. 
 ~P X  |  A. b  e.  a  E  e.  a }  e.  (ACS `  X ) )
 
Theoremacsfn1c 13512* Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( X  e.  V  /\  A. b  e.  K  A. c  e.  X  E  e.  X )  ->  { a  e. 
 ~P X  |  A. b  e.  K  A. c  e.  a  E  e.  a }  e.  (ACS `  X ) )
 
Theoremacsfn2 13513* Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( X  e.  V  /\  A. b  e.  X  A. c  e.  X  E  e.  X )  ->  { a  e. 
 ~P X  |  A. b  e.  a  A. c  e.  a  E  e.  a }  e.  (ACS `  X ) )
 
PART 8  BASIC CATEGORY THEORY
 
8.1  Categories
 
8.1.1  Categories
 
Syntaxccat 13514 Extend class notation with the class of categories.
 class  Cat
 
Syntaxccid 13515 Extend class notation with the identity arrow of a category.
 class  Id
 
Syntaxchomf 13516 Extend class notation to include functionalized Hom-set extractor.
 class  Homf
 
Syntaxccomf 13517 Extend class notation to include functionalized composition operation.
 class compf
 
Definitiondf-cat 13518* A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated to those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
 |- 
 Cat  =  { c  |  [. ( Base `  c
 )  /  b ]. [. (  Hom  `  c
 )  /  h ]. [. (comp `  c )  /  o ]. A. x  e.  b  ( E. g  e.  ( x h x ) A. y  e.  b  ( A. f  e.  (
 y h x ) ( g ( <. y ,  x >. o x ) f )  =  f  /\  A. f  e.  ( x h y ) ( f (
 <. x ,  x >. o y ) g )  =  f )  /\  A. y  e.  b  A. z  e.  b  A. f  e.  ( x h y ) A. g  e.  ( y h z ) ( ( g ( <. x ,  y >. o z ) f )  e.  ( x h z )  /\  A. w  e.  b  A. k  e.  ( z h w ) ( ( k ( <. y ,  z >. o w ) g ) ( <. x ,  y >. o w ) f )  =  ( k ( <. x ,  z >. o w ) ( g ( <. x ,  y >. o z ) f ) ) ) ) }
 
Definitiondf-cid 13519* Define the category identity arrow. Since it is uniquely defined when it exists, we do not need to add it to the data of the category, and instead extract it by uniqueness. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- 
 Id  =  ( c  e.  Cat  |->  [_ ( Base `  c )  /  b ]_ [_ (  Hom  `  c )  /  h ]_
 [_ (comp `  c
 )  /  o ]_ ( x  e.  b  |->  ( iota_ g  e.  ( x h x ) A. y  e.  b  ( A. f  e.  (
 y h x ) ( g ( <. y ,  x >. o x ) f )  =  f  /\  A. f  e.  ( x h y ) ( f (
 <. x ,  x >. o y ) g )  =  f ) ) ) )
 
Definitiondf-homf 13520* Define the functionalized Hom-set operator, which is exactly like  Hom but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |- 
 Homf  =  ( c  e.  _V  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c )  |->  ( x (  Hom  `  c
 ) y ) ) )
 
Definitiondf-comf 13521* Define the functionalized composition operator, which is exactly like comp but is guaranteed to be a function of the proper type. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |- compf  =  ( c  e.  _V  |->  ( x  e.  (
 ( Base `  c )  X.  ( Base `  c )
 ) ,  y  e.  ( Base `  c )  |->  ( g  e.  (
 ( 2nd `  x )
 (  Hom  `  c ) y ) ,  f  e.  ( (  Hom  `  c
 ) `  x )  |->  ( g ( x (comp `  c )
 y ) f ) ) ) )
 
Theoremiscat 13522* The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   =>    |-  ( C  e.  V  ->  ( C  e.  Cat  <->  A. x  e.  B  ( E. g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  (
 y H x ) ( g ( <. y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f (
 <. x ,  x >.  .x.  y ) g )  =  f )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( ( g ( <. x ,  y >.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <. y ,  z >.  .x.  w ) g ) ( <. x ,  y >.  .x.  w ) f )  =  ( k ( <. x ,  z >.  .x.  w ) ( g (
 <. x ,  y >.  .x.  z ) f ) ) ) ) ) )
 
Theoremiscatd 13523* Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  B  =  ( Base `  C )
 )   &    |-  ( ph  ->  H  =  (  Hom  `  C ) )   &    |-  ( ph  ->  .x. 
 =  (comp `  C ) )   &    |-  ( ph  ->  C  e.  V )   &    |-  (
 ( ph  /\  x  e.  B )  ->  .1.  e.  ( x H x ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  f  e.  (
 y H x ) ) )  ->  (  .1.  ( <. y ,  x >.  .x.  x ) f )  =  f )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  f  e.  ( x H y ) ) )  ->  ( f
 ( <. x ,  x >.  .x.  y )  .1.  )  =  f )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  /\  ( f  e.  ( x H y )  /\  g  e.  ( y H z ) ) )  ->  ( g
 ( <. x ,  y >.  .x.  z ) f )  e.  ( x H z ) )   &    |-  ( ( ph  /\  (
 ( x  e.  B  /\  y  e.  B )  /\  ( z  e.  B  /\  w  e.  B ) )  /\  ( f  e.  ( x H y )  /\  g  e.  ( y H z )  /\  k  e.  ( z H w ) ) ) 
 ->  ( ( k (
 <. y ,  z >.  .x. 
 w ) g ) ( <. x ,  y >.  .x.  w ) f )  =  ( k ( <. x ,  z >.  .x.  w ) ( g ( <. x ,  y >.  .x.  z )
 f ) ) )   =>    |-  ( ph  ->  C  e.  Cat )
 
Theoremcatidex 13524* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  E. g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  (
 y H X ) ( g ( <. y ,  X >.  .x.  X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f (
 <. X ,  X >.  .x.  y ) g )  =  f ) )
 
Theoremcatideu 13525* Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  E! g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  ( y H X ) ( g (
 <. y ,  X >.  .x. 
 X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f ( <. X ,  X >.  .x.  y )
 g )  =  f ) )
 
Theoremcidfval 13526* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |- 
 .1.  =  ( Id `  C )   =>    |-  ( ph  ->  .1.  =  ( x  e.  B  |->  ( iota_ g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  (
 y H x ) ( g ( <. y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f (
 <. x ,  x >.  .x.  y ) g )  =  f ) ) ) )
 
Theoremcidval 13527* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |- 
 .1.  =  ( Id `  C )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (  .1.  `  X )  =  ( iota_ g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  (
 y H X ) ( g ( <. y ,  X >.  .x.  X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f (
 <. X ,  X >.  .x.  y ) g )  =  f ) ) )
 
Theoremcidffn 13528 The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |- 
 Id  Fn  Cat
 
Theoremcidfn 13529 The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  .1.  =  ( Id `  C )   =>    |-  ( C  e.  Cat  ->  .1.  Fn  B )
 
Theoremcatidd 13530* Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  B  =  ( Base `  C )
 )   &    |-  ( ph  ->  H  =  (  Hom  `  C ) )   &    |-  ( ph  ->  .x. 
 =  (comp `  C ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  (
 ( ph  /\  x  e.  B )  ->  .1.  e.  ( x H x ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  f  e.  (
 y H x ) ) )  ->  (  .1.  ( <. y ,  x >.  .x.  x ) f )  =  f )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  f  e.  ( x H y ) ) )  ->  ( f
 ( <. x ,  x >.  .x.  y )  .1.  )  =  f )   =>    |-  ( ph  ->  ( Id `  C )  =  ( x  e.  B  |->  .1.  ) )
 
Theoremiscatd2 13531* Version of iscatd2 13531 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  B  =  ( Base `  C )
 )   &    |-  ( ph  ->  H  =  (  Hom  `  C ) )   &    |-  ( ph  ->  .x. 
 =  (comp `  C ) )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ps 
 <->  ( ( x  e.  B  /\  y  e.  B )  /\  (
 z  e.  B  /\  w  e.  B )  /\  ( f  e.  ( x H y )  /\  g  e.  ( y H z )  /\  k  e.  ( z H w ) ) ) )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  .1.  e.  ( y H y ) )   &    |-  ( ( ph  /\  ps )  ->  (  .1.  ( <. x ,  y >.  .x.  y ) f )  =  f )   &    |-  (
 ( ph  /\  ps )  ->  ( g ( <. y ,  y >.  .x.  z
 )  .1.  )  =  g )   &    |-  ( ( ph  /\ 
 ps )  ->  (
 g ( <. x ,  y >.  .x.  z )
 f )  e.  ( x H z ) )   &    |-  ( ( ph  /\  ps )  ->  ( ( k ( <. y ,  z >.  .x.  w ) g ) ( <. x ,  y >.  .x.  w )
 f )  =  ( k ( <. x ,  z >.  .x.  w )
 ( g ( <. x ,  y >.  .x.  z
 ) f ) ) )   =>    |-  ( ph  ->  ( C  e.  Cat  /\  ( Id `  C )  =  ( y  e.  B  |->  .1.  ) ) )
 
Theoremcatidcl 13532 Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (  .1.  `  X )  e.  ( X H X ) )
 
Theoremcatlid 13533 Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   =>    |-  ( ph  ->  (
 (  .1.  `  Y ) ( <. X ,  Y >.  .x.  Y ) F )  =  F )
 
Theoremcatrid 13534 Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( F ( <. X ,  X >.  .x.  Y )
 (  .1.  `  X ) )  =  F )
 
Theoremcatcocl 13535 Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X H Z ) )
 
Theoremcatass 13536 Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   &    |-  ( ph  ->  W  e.  B )   &    |-  ( ph  ->  K  e.  ( Z H W ) )   =>    |-  ( ph  ->  (
 ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W )
 ( G ( <. X ,  Y >.  .x.  Z ) F ) ) )
 
Theorem0catg 13537 Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  ->  C  e.  Cat )
 
Theorem0cat 13538 The empty set is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  (/)  e.  Cat
 
Theoremproplem2 13539* Lemma for mndpropd 14346. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  ( ( ( X  e.  A  /\  Y  e.  B )  /\  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C )  ->  ( X F Y )  e.  C )
 
Theoremproplem 13540* Lemma for mndpropd 14346. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ( x F y )  =  ( x G y ) )   =>    |-  ( ( ph  /\  ( X  e.  A  /\  Y  e.  B )
 )  ->  ( X F Y )  =  ( X G Y ) )
 
Theoremproplem3 13541 Lemma for property theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ( ph  /\  ps )  ->  ( x F y )  =  ( x G y ) )
 
Theoremhomffval 13542* Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  F  =  (  Homf  `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   =>    |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )
 
Theoremhomfval 13543 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  F  =  (  Homf  `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X F Y )  =  ( X H Y ) )
 
Theoremhomffn 13544 The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  F  =  (  Homf  `  C )   &    |-  B  =  ( Base `  C )   =>    |-  F  Fn  ( B  X.  B )
 
Theoremhomfeq 13545* Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  H  =  (  Hom  `  C )   &    |-  J  =  ( 
 Hom  `  D )   &    |-  ( ph  ->  B  =  (
 Base `  C ) )   &    |-  ( ph  ->  B  =  ( Base `  D )
 )   =>    |-  ( ph  ->  (
 (  Homf  `  C )  =  ( 
 Homf  `  D )  <->  A. x  e.  B  A. y  e.  B  ( x H y )  =  ( x J y ) ) )
 
Theoremhomfeqd 13546 If two structures have the same 
Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ( Base `  C )  =  ( Base `  D )
 )   &    |-  ( ph  ->  (  Hom  `  C )  =  (  Hom  `  D ) )   =>    |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )
 
Theoremhomfeqbas 13547 Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )   =>    |-  ( ph  ->  (
 Base `  C )  =  ( Base `  D )
 )
 
Theoremhomfeqval 13548 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X H Y )  =  ( X J Y ) )
 
Theoremcomfffval 13549* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   =>    |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `
  x )  |->  ( g ( x  .x.  y ) f ) ) )
 
Theoremcomffval 13550* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y )  |->  ( g ( <. X ,  Y >.  .x.  Z )
 f ) ) )
 
Theoremcomfval 13551 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  ( G ( <. X ,  Y >.  .x.  Z ) F ) )
 
Theoremcomfffval2 13552* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Homf  `  C )   &    |-  .x.  =  (comp `  C )   =>    |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `
  x )  |->  ( g ( x  .x.  y ) f ) ) )
 
Theoremcomffval2 13553* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Homf  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y )  |->  ( g ( <. X ,  Y >.  .x.  Z )
 f ) ) )
 
Theoremcomfval2 13554 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Homf  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  ( G ( <. X ,  Y >.  .x.  Z ) F ) )
 
Theoremcomfffn 13555 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   =>    |-  O  Fn  ( ( B  X.  B )  X.  B )
 
Theoremcomffn 13556 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 <. X ,  Y >. O Z )  Fn  (
 ( Y H Z )  X.  ( X H Y ) ) )
 
Theoremcomfeq 13557* Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |- 
 .x.  =  (comp `  C )   &    |-  .xb  =  (comp `  D )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  B  =  ( Base `  C ) )   &    |-  ( ph  ->  B  =  ( Base `  D ) )   &    |-  ( ph  ->  ( 
 Homf  `  C )  =  ( 
 Homf  `  D ) )   =>    |-  ( ph  ->  ( (compf `  C )  =  (compf `  D )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( g (
 <. x ,  y >.  .x.  z ) f )  =  ( g (
 <. x ,  y >.  .xb  z ) f ) ) )
 
Theoremcomfeqd 13558 Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  (comp `  C )  =  (comp `  D ) )   &    |-  ( ph  ->  (  Homf  `  C )  =  (  Homf  `  D ) )   =>    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
 
Theoremcomfeqval 13559 Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  .xb  =  (comp `  D )   &    |-  ( ph  ->  (  Homf  `  C )  =  (  Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  =  ( G ( <. X ,  Y >. 
 .xb  Z ) F ) )
 
Theoremcatpropd 13560 Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  W )   =>    |-  ( ph  ->  ( C  e.  Cat  <->  D  e.  Cat ) )
 
Theoremcidpropd 13561 Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  W )   =>    |-  ( ph  ->  ( Id `  C )  =  ( Id `  D ) )
 
8.1.2  Opposite category
 
Syntaxcoppc 13562 The opposite category operation.
 class oppCat
 
Definitiondf-oppc 13563* Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- oppCat  =  ( f  e.  _V  |->  ( ( f sSet  <. ( 
 Hom  `  ndx ) , tpos 
 (  Hom  `  f )
 >. ) sSet  <. (comp `  ndx ) ,  ( u  e.  ( ( Base `  f )  X.  ( Base `  f ) ) ,  z  e.  ( Base `  f )  |-> tpos  ( <. z ,  ( 2nd `  u ) >. (comp `  f ) ( 1st `  u ) ) )
 >. ) )
 
Theoremoppcval 13564* Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  O  =  (oppCat `  C )   =>    |-  ( C  e.  V  ->  O  =  ( ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. ) sSet  <. (comp `  ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) ) ) >. ) )
 
Theoremoppchomfval 13565 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  H  =  (  Hom  `  C )   &    |-  O  =  (oppCat `  C )   =>    |- tpos  H  =  (  Hom  `  O )
 
Theoremoppchom 13566 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  H  =  (  Hom  `  C )   &    |-  O  =  (oppCat `  C )   =>    |-  ( X (  Hom  `  O ) Y )  =  ( Y H X )
 
Theoremoppccofval 13567 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  .x.  =  (comp `  C )   &    |-  O  =  (oppCat `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( <. X ,  Y >. (comp `  O ) Z )  = tpos  ( <. Z ,  Y >.  .x.  X )
 )
 
Theoremoppcco 13568 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  .x.  =  (comp `  C )   &    |-  O  =  (oppCat `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  O ) Z ) F )  =  ( F (
 <. Z ,  Y >.  .x. 
 X ) G ) )
 
Theoremoppcbas 13569 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  B  =  ( Base `  C )   =>    |-  B  =  ( Base `  O )
 
Theoremoppccatid 13570 Lemma for oppccat 13573. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  ( C  e.  Cat  ->  ( O  e.  Cat  /\  ( Id `  O )  =  ( Id `  C ) ) )
 
Theoremoppchomf 13571 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  H  =  (  Homf  `  C )   =>    |- tpos  H  =  (  Homf  `  O )
 
Theoremoppcid 13572 Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  B  =  ( Id
 `  C )   =>    |-  ( C  e.  Cat 
 ->  ( Id `  O )  =  B )
 
Theoremoppccat 13573 An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  ( C  e.  Cat  ->  O  e.  Cat )
 
Theorem2oppcbas 13574 The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 13588. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  B  =  ( Base `  C )   =>    |-  B  =  ( Base `  (oppCat `  O )
 )
 
Theorem2oppchomf 13575 The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 13588. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  (  Homf  `  C )  =  ( 
 Homf  `  (oppCat `  O )
 )
 
Theorem2oppccomf 13576 The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 13588. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  (compf `  C )  =  (compf `  (oppCat `  O ) )
 
Theoremoppchomfpropd 13577 If two categories have the same hom-sets, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )   =>    |-  ( ph  ->  ( 
 Homf  `  (oppCat `  C )
 )  =  (  Homf  `  (oppCat `  D ) ) )
 
Theoremoppccomfpropd 13578 If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   =>    |-  ( ph  ->  (compf `  (oppCat `  C ) )  =  (compf `  (oppCat `  D )
 ) )
 
8.1.3  Monomorphisms and epimorphisms
 
Syntaxcmon 13579 Extend class notation with the class of all monomorphisms.
 class Mono
 
Syntaxcepi 13580 Extend class notation with the class of all epimorphisms.
 class Epi
 
Definitiondf-mon 13581* Function returning the monomorphisms of the category  c. JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
 |- Mono  =  ( c  e.  Cat  |->  [_ ( Base `  c )  /  b ]_ [_ (  Hom  `  c )  /  h ]_ ( x  e.  b ,  y  e.  b  |->  { f  e.  ( x h y )  | 
 A. z  e.  b  Fun  `' ( g  e.  (
 z h x ) 
 |->  ( f ( <. z ,  x >. (comp `  c ) y ) g ) ) }
 ) )
 
Definitiondf-epi 13582 Function returning the epimorphisms of the category  c. JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
 |- Epi 
 =  ( c  e. 
 Cat  |-> tpos  (Mono `  (oppCat `  c
 ) ) )
 
Theoremmonfval 13583* Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  M  =  (Mono `  C )   &    |-  ( ph  ->  C  e.  Cat )   =>    |-  ( ph  ->  M  =  ( x  e.  B ,  y  e.  B  |->  { f  e.  ( x H y )  | 
 A. z  e.  B  Fun  `' ( g  e.  (
 z H x ) 
 |->  ( f ( <. z ,  x >.  .x.  y
 ) g ) ) } ) )
 
Theoremismon 13584* Definition of a monomorphism in a category. (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  ->  ( F  e.  ( X M Y )  <->  ( F  e.  ( X H Y ) 
 /\  A. z  e.  B  Fun  `' ( g  e.  (
 z H X ) 
 |->  ( F ( <. z ,  X >.  .x.  Y ) g ) ) ) ) )
 
Theoremismon2 13585* Write out the monomorphism property directly. (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  ->  ( F  e.  ( X M Y )  <->  ( F  e.  ( X H Y ) 
 /\  A. z  e.  B  A. g  e.  ( z H X ) A. h  e.  ( z H X ) ( ( F ( <. z ,  X >.  .x.  Y ) g )  =  ( F ( <. z ,  X >.  .x.  Y ) h )  ->  g  =  h ) ) ) )
 
Theoremmonhom 13586 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 13587 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 13588 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 13589 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 13590 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 13591* 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 13592* 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 13593 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 13594 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 13595 Extend class notation with the sections of a morphism.
 class Sect
 
Syntaxcinv 13596 Extend class notation with the inverses of a morphism.
 class Inv
 
Syntaxciso 13597 Extend class notation with the class of all isomorphisms.
 class  Iso
 
Definitiondf-sect 13598* 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 13599* 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 13600* 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
 ) ) )
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