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Theorem List for Metamath Proof Explorer - 13601-13700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcomfval2 13601 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 13602 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 13603 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 13604* 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 13605 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 13606 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 13607 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 13608 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 13609 The opposite category operation.
 class oppCat
 
Definitiondf-oppc 13610* 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 13611* 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 13612 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  H  =  (  Hom  `  C )   &    |-  O  =  (oppCat `  C )   =>    |- tpos  H  =  (  Hom  `  O )
 
Theoremoppchom 13613 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 13614 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 13615 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 13616 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  B  =  ( Base `  C )   =>    |-  B  =  ( Base `  O )
 
Theoremoppccatid 13617 Lemma for oppccat 13620. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  ( C  e.  Cat  ->  ( O  e.  Cat  /\  ( Id `  O )  =  ( Id `  C ) ) )
 
Theoremoppchomf 13618 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  H  =  (  Homf  `  C )   =>    |- tpos  H  =  (  Homf  `  O )
 
Theoremoppcid 13619 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 13620 An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  ( C  e.  Cat  ->  O  e.  Cat )
 
Theorem2oppcbas 13621 The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 13635. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  B  =  ( Base `  C )   =>    |-  B  =  ( Base `  (oppCat `  O )
 )
 
Theorem2oppchomf 13622 The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 13635. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  (  Homf  `  C )  =  ( 
 Homf  `  (oppCat `  O )
 )
 
Theorem2oppccomf 13623 The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 13635. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  O  =  (oppCat `  C )   =>    |-  (compf `  C )  =  (compf `  (oppCat `  O ) )
 
Theoremoppchomfpropd 13624 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 13625 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 13626 Extend class notation with the class of all monomorphisms.
 class Mono
 
Syntaxcepi 13627 Extend class notation with the class of all epimorphisms.
 class Epi
 
Definitiondf-mon 13628* 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 13629 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 13630* 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 13631* 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 13632* 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 13633 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 13634 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 13635 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 13636 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 13637 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 13638* 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 13639* 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 13640 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 13641 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 13642 Extend class notation with the sections of a morphism.
 class Sect
 
Syntaxcinv 13643 Extend class notation with the inverses of a morphism.
 class Inv
 
Syntaxciso 13644 Extend class notation with the class of all isomorphisms.
 class  Iso
 
Definitiondf-sect 13645* 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 13646* 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 13647* 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 13648* 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 13649* 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 13650 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 13651 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 13652 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 13653 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 13654 Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  .x.  =  (comp `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F ( X S Y ) G )   &    |-  ( ph  ->  H ( Y S Z ) K )   =>    |-  ( ph  ->  ( H ( <. X ,  Y >.  .x.  Z ) F ) ( X S Z ) ( G ( <. Z ,  Y >.  .x.  X ) K ) )
 
Theoreminvffval 13655* 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 13656 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 13657 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 13658 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 13659 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 13660 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 13661 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 13662 The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   =>    |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
 
Theoreminviso1 13663 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 13664 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 13665 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 13666 The inverse relation is a bijection from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   =>    |-  ( ph  ->  ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )
 
Theoreminvinv 13667 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 13668 The composition of two isomorphisms is an isomorphism, and the inverse is the composition of the individual inverses. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  ( ph  ->  F  e.  ( X I Y ) )   &    |-  .x. 
 =  (comp `  C )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  G  e.  ( Y I Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F ) ( X N Z ) ( ( ( X N Y ) `  F ) ( <. Z ,  Y >.  .x.  X )
 ( ( Y N Z ) `  G ) ) )
 
Theoremisohom 13669 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 13670 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 13671 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 13672 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 13673 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 13674 An isomorphism in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  O  =  (oppCat `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  ( 
 Iso  `  C )   &    |-  J  =  (  Iso  `  O )   =>    |-  ( ph  ->  ( X J Y )  =  ( Y I X ) )
 
Theoremsectmon 13675 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 13676 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 13677 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 13678 If  F is an epimorphism and  F is a section of  G, then  G is an inverse of  F and they are both isomorphisms. This is also stated as "a epimorphism which is also a split monomorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  E  =  (Epi `  C )   &    |-  S  =  (Sect `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  N  =  (Inv `  C )   &    |-  ( ph  ->  F  e.  ( X E Y ) )   &    |-  ( ph  ->  F ( X S Y ) G )   =>    |-  ( ph  ->  F ( X N Y ) G )
 
8.1.5  Subcategories
 
Syntaxcssc 13679 Extend class notation to include the subset relation for subcategories.
 class  C_cat
 
Syntaxcresc 13680 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
 class  |`cat
 
Syntaxcsubc 13681 Extend class notation to include the collection of subcategories of a category.
 class Subcat
 
Definitiondf-ssc 13682* 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 13684, 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 13683* 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 13684*  (Subcat `  C
) is the set of all the subcategory specifications of the category  C. Like df-subg 14613, this is not actually a collection of categories, but only sets which when given operations from the base category (using df-resc 13683) form a category. All the objects and all the morphisms of the subcategory belong to the supercategory. The identity of an object, the domain and the codomain of a morphism are the same in the subcategory and the supercategory. The composition of the subcategory is a restriction of the composition of the supercategory. (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 4-Jan-2017.)
 |- Subcat  =  ( c  e.  Cat  |->  { h  |  ( h 
 C_cat  (  Homf  `  c )  /\  [. dom  dom 
 h  /  s ]. A. x  e.  s  ( ( ( Id `  c ) `  x )  e.  ( x h x )  /\  A. y  e.  s  A. z  e.  s  A. f  e.  ( x h y ) A. g  e.  ( y h z ) ( g ( <. x ,  y >. (comp `  c
 ) z ) f )  e.  ( x h z ) ) ) } )
 
Theoremsscrel 13685 The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |- 
 Rel  C_cat
 
Theorembrssc 13686* 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 13687* An analogue of pwex 4191 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 13688 Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( H  e.  (Subcat `  C )  ->  C  e.  Cat )
 
Theoremsscfn1 13689 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 13690 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 13691 Lemma for ssc1 13693 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  H  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( H  e.  _V  <->  S  e.  _V ) )
 
Theoremisssc 13692* 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 13693 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 13694 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 13695 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 13696 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 13697 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 13698 The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ( A  C_cat  B  /\  B  C_cat  A )  ->  A  =  B )
 
Theoremrescval 13699 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 13700 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 >. ) )
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