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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremacsfiel 13401* 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 13402* 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 )
 )
 
Theoremisacs1i 13403* 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 13404 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 13405* 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 13406* 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 13407* 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 13408* 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 13409* 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 13410 Extend class notation with the class of categories.
 class  Cat
 
Syntaxccid 13411 Extend class notation with the identity arrow of a category.
 class  Id
 
Syntaxchomf 13412 Extend class notation to include functionalized Hom-set extractor.
 class  Homf
 
Syntaxccomf 13413 Extend class notation to include functionalized composition operation.
 class compf
 
Definitiondf-cat 13414* 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 13415* 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 13416* 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 13417* 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 13418* 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 13419* 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 13420* 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 13421* 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 13422* 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 13423* 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 13424 The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |- 
 Id  Fn  Cat
 
Theoremcidfn 13425 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 13426* 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 13427* Version of iscatd2 13427 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 13428 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 13429 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 13430 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 13431 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 13432 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 13433 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 13434 The empty set is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  (/)  e.  Cat
 
Theoremproplem2 13435* Lemma for mndpropd 14233. (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 13436* Lemma for mndpropd 14233. (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 13437 Lemma for property theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ( ph  /\  ps )  ->  ( x F y )  =  ( x G y ) )
 
Theoremhomffval 13438* 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 13439 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 13440 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 13441* 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 13442 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 13443 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 13444 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 13445* 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 13446* 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 13447 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 13448* 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 13449* 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 13450 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 13451 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 13452 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 13453* 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 13454 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 13455 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 13456 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 13457 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 13458 The opposite category operation.
 class oppCat
 
Definitiondf-oppc 13459* 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 13460* 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 13461 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  H  =  (  Hom  `  C )   &    |-  O  =  (oppCat `  C )   =>    |- tpos  H  =  (  Hom  `  O )
 
Theoremoppchom 13462 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 13463 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 13464 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 13465 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  B  =  ( Base `  C )   =>    |-  B  =  ( Base `  O )
 
Theoremoppccatid 13466 Lemma for oppccat 13469. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  ( C  e.  Cat  ->  ( O  e.  Cat  /\  ( Id `  O )  =  ( Id `  C ) ) )
 
Theoremoppchomf 13467 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  H  =  (  Homf  `  C )   =>    |- tpos  H  =  (  Homf  `  O )
 
Theoremoppcid 13468 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 13469 An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  ( C  e.  Cat  ->  O  e.  Cat )
 
Theorem2oppcbas 13470 The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 13484. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  B  =  ( Base `  C )   =>    |-  B  =  ( Base `  (oppCat `  O )
 )
 
Theorem2oppchomf 13471 The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 13484. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  (  Homf  `  C )  =  ( 
 Homf  `  (oppCat `  O )
 )
 
Theorem2oppccomf 13472 The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 13484. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  (compf `  C )  =  (compf `  (oppCat `  O ) )
 
Theoremoppchomfpropd 13473 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 13474 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 13475 Extend class notation with the class of all monomorphisms.
 class Mono
 
Syntaxcepi 13476 Extend class notation with the class of all epimorphisms.
 class Epi
 
Definitiondf-mon 13477* 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 13478 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 13479* 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 13480* 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 13481* 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 13482 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 13483 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 13484 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 13485 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 13486 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 13487* 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 13488* 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 13489 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 13490 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 13491 Extend class notation with the sections of a morphism.
 class Sect
 
Syntaxcinv 13492 Extend class notation with the inverses of a morphism.
 class Inv
 
Syntaxciso 13493 Extend class notation with the class of all isomorphisms.
 class  Iso
 
Definitiondf-sect 13494* 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 13495* 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 13496* 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 13497* 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 13498* 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 13499 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 13500 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 ) ) ) )
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