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Theorem List for Metamath Proof Explorer - 13501-13600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremissect2 13501 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 13502 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 13503 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 13504* 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 13505 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 13506 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 13507 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 13508 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 13509 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 13510 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 13511 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 13512 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 13513 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 13514 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 13515 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 13516 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 13517 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 13518 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 13519 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 13520 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 13521 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 13522 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 13523 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 13524 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 13525 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 13526 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 13527 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 13528 Extend class notation to include the subset relation for subcategories.
 class  C_cat
 
Syntaxcresc 13529 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
 class  |`cat
 
Syntaxcsubc 13530 Extend class notation to include the collection of subcategories of a category.
 class Subcat
 
Definitiondf-ssc 13531* 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 13533, 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 13532* 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 13533*  (Subcat `  C
) is the set of all the subcategory specifications of the category  C. Like df-subg 14453, this is not actually a collection of categories, but only sets which when given operations from the base category (using df-resc 13532) 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 13534 The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |- 
 Rel  C_cat
 
Theorembrssc 13535* 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 13536* An analogue of pwex 4087 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 13537 Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( H  e.  (Subcat `  C )  ->  C  e.  Cat )
 
Theoremsscfn1 13538 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 13539 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 13540 Lemma for ssc1 13542 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  H  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( H  e.  _V  <->  S  e.  _V ) )
 
Theoremisssc 13541* 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 13542 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 13543 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 13544 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 13545 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 13546 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 13547 The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ( A  C_cat  B  /\  B  C_cat  A )  ->  A  =  B )
 
Theoremrescval 13548 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 13549 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  D  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
 
Theoremrescbas 13550 Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   =>    |-  ( ph  ->  S  =  ( Base `  D ) )
 
Theoremreschom 13551 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   =>    |-  ( ph  ->  H  =  (  Hom  `  D ) )
 
Theoremreschomf 13552 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   =>    |-  ( ph  ->  H  =  (  Homf  `  D ) )
 
Theoremrescco 13553 Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   &    |-  B  =  (
 Base `  C )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  S 
 C_  B )   &    |-  .x.  =  (comp `  C )   =>    |-  ( ph  ->  .x. 
 =  (comp `  D ) )
 
Theoremrescabs 13554 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  J  Fn  ( T  X.  T ) )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  T 
 C_  S )   =>    |-  ( ph  ->  ( ( C  |`cat  H )  |`cat  J )  =  ( C  |`cat  J ) )
 
Theoremrescabs2 13555 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  J  Fn  ( T  X.  T ) )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  T  C_  S )   =>    |-  ( ph  ->  (
 ( Cs  S )  |`cat  J )  =  ( C  |`cat  J )
 )
 
Theoremissubc 13556* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  H  =  (  Homf  `  C )   &    |- 
 .1.  =  ( Id `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  S  =  dom  dom 
 J )   =>    |-  ( ph  ->  ( J  e.  (Subcat `  C ) 
 <->  ( J  C_cat  H  /\  A. x  e.  S  ( (  .1.  `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
 <. x ,  y >.  .x.  z ) f )  e.  ( x J z ) ) ) ) )
 
Theoremissubc2 13557* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  H  =  (  Homf  `  C )   &    |- 
 .1.  =  ( Id `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  H  /\  A. x  e.  S  ( (  .1.  `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
 <. x ,  y >.  .x.  z ) f )  e.  ( x J z ) ) ) ) )
 
Theoremsubcssc 13558 An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  H  =  (  Homf  `  C )   =>    |-  ( ph  ->  J  C_cat  H )
 
Theoremsubcfn 13559 An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  S  =  dom  dom  J )   =>    |-  ( ph  ->  J  Fn  ( S  X.  S ) )
 
Theoremsubcss1 13560 The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  B  =  (
 Base `  C )   =>    |-  ( ph  ->  S 
 C_  B )
 
Theoremsubcss2 13561 The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  ( X J Y )  C_  ( X H Y ) )
 
Theoremsubcidcl 13562 The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  .1.  =  ( Id `  C )   =>    |-  ( ph  ->  (  .1.  `  X )  e.  ( X J X ) )
 
Theoremsubccocl 13563 A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  Y  e.  S )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ph  ->  F  e.  ( X J Y ) )   &    |-  ( ph  ->  G  e.  ( Y J Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) )
 
Theoremsubccatid 13564* A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  .1.  =  ( Id `  C )   =>    |-  ( ph  ->  ( D  e.  Cat  /\  ( Id `  D )  =  ( x  e.  S  |->  (  .1.  `  x )
 ) ) )
 
Theoremsubcid 13565 The identity in a subcategory is the same as the original category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  X  e.  S )   =>    |-  ( ph  ->  (  .1.  `  X )  =  ( ( Id `  D ) `  X ) )
 
Theoremsubccat 13566 A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   =>    |-  ( ph  ->  D  e.  Cat )
 
Theoremissubc3 13567* Alternate definition of a subcategory, as a subset of the category which is itself a category. The assumption that the identity be closed is necessary just as in the case of a monoid, issubm2 14261, for the same reasons, since categories are a generalization of monoids. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  H  =  (  Homf  `  C )   &    |- 
 .1.  =  ( Id `  C )   &    |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  H  /\  A. x  e.  S  (  .1.  `  x )  e.  ( x J x )  /\  D  e.  Cat ) ) )
 
Theoremfullsubc 13568 The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory). (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Homf  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  S  C_  B )   =>    |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  e.  (Subcat `  C )
 )
 
Theoremfullresc 13569 The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Homf  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  S  C_  B )   &    |-  D  =  ( Cs  S )   &    |-  E  =  ( C  |`cat  ( H  |`  ( S  X.  S ) ) )   =>    |-  ( ph  ->  (
 (  Homf  `  D )  =  ( 
 Homf  `  E )  /\  (compf `  D )  =  (compf `  E ) ) )
 
Theoremresscat 13570 A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ( C  e.  Cat  /\  S  e.  V ) 
 ->  ( Cs  S )  e.  Cat )
 
Theoremsubsubc 13571 A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   =>    |-  ( H  e.  (Subcat `  C )  ->  ( J  e.  (Subcat `  D ) 
 <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H ) ) )
 
8.1.6  Functors
 
Syntaxcfunc 13572 Extend class notation with the class of all functors.
 class  Func
 
Syntaxcidfu 13573 Extend class notation with identity functor.
 class idfunc
 
Syntaxccofu 13574 Extend class notation with functor composition.
 class  o.func
 
Syntaxcresf 13575 Extend class notation to include restriction of a functor to a subcategory.
 class  |`f
 
Definitiondf-func 13576* Function returning all the functors from a category  t to a category  u. Intuitively a functor associates any morphism of  t to a morphism of  u, any object of  t to an object of  u, and respects the identity, the composition, the domain and the codomain. Here to capture the idea that a functor associates any object of  t to an object of  u we write it associates any identity of  t to an identity of  u which simplifies the definition. (Contributed by FL, 10-Feb-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
 |- 
 Func  =  ( t  e.  Cat ,  u  e. 
 Cat  |->  { <. f ,  g >.  |  [. ( Base `  t )  /  b ]. ( f : b --> ( Base `  u )  /\  g  e.  X_ z  e.  ( b  X.  b
 ) ( ( ( f `  ( 1st `  z ) ) ( 
 Hom  `  u )
 ( f `  ( 2nd `  z ) ) )  ^m  ( ( 
 Hom  `  t ) `  z ) )  /\  A. x  e.  b  ( ( ( x g x ) `  (
 ( Id `  t
 ) `  x )
 )  =  ( ( Id `  u ) `
  ( f `  x ) )  /\  A. y  e.  b  A. z  e.  b  A. m  e.  ( x (  Hom  `  t )
 y ) A. n  e.  ( y (  Hom  `  t ) z ) ( ( x g z ) `  ( n ( <. x ,  y >. (comp `  t
 ) z ) m ) )  =  ( ( ( y g z ) `  n ) ( <. ( f `
  x ) ,  ( f `  y
 ) >. (comp `  u ) ( f `  z ) ) ( ( x g y ) `  m ) ) ) ) }
 )
 
Definitiondf-idfu 13577* Define the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- idfunc  =  ( t  e.  Cat  |->  [_ ( Base `  t )  /  b ]_ <. (  _I  |`  b ) ,  (
 z  e.  ( b  X.  b )  |->  (  _I  |`  ( (  Hom  `  t ) `  z ) ) )
 >. )
 
Definitiondf-cofu 13578* Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- 
 o.func  =  ( g  e.  _V ,  f  e.  _V  |->  <.
 ( ( 1st `  g
 )  o.  ( 1st `  f ) ) ,  ( x  e.  dom  dom  ( 2nd `  f
 ) ,  y  e. 
 dom  dom  ( 2nd `  f
 )  |->  ( ( ( ( 1st `  f
 ) `  x )
 ( 2nd `  g )
 ( ( 1st `  f
 ) `  y )
 )  o.  ( x ( 2nd `  f
 ) y ) ) ) >. )
 
Definitiondf-resf 13579* Define the restriction of a functor to a subcategory (analogue of df-res 4600). (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  |`f  =  ( f  e.  _V ,  h  e.  _V  |->  <.
 ( ( 1st `  f
 )  |`  dom  dom  h ) ,  ( x  e. 
 dom  h  |->  ( ( ( 2nd `  f
 ) `  x )  |`  ( h `  x ) ) ) >. )
 
Theoremrelfunc 13580 The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 Rel  ( D  Func  E )
 
Theoremfuncrcl 13581 Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( F  e.  ( D  Func  E )  ->  ( D  e.  Cat  /\  E  e.  Cat )
 )
 
Theoremisfunc 13582* Value of the set of functors between two categories. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  C  =  (
 Base `  E )   &    |-  H  =  (  Hom  `  D )   &    |-  J  =  (  Hom  `  E )   &    |-  .1.  =  ( Id `  D )   &    |-  I  =  ( Id `  E )   &    |-  .x.  =  (comp `  D )   &    |-  O  =  (comp `  E )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   =>    |-  ( ph  ->  ( F ( D  Func  E ) G  <->  ( F : B
 --> C  /\  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `  ( 1st `  z )
 ) J ( F `
  ( 2nd `  z
 ) ) )  ^m  ( H `  z ) )  /\  A. x  e.  B  ( ( ( x G x ) `
  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x H y ) A. n  e.  ( y H z ) ( ( x G z ) `  ( n ( <. x ,  y >.  .x.  z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) ) ) ) ) )
 
Theoremisfuncd 13583* Deduce that an operation is a functor of categories. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  C  =  (
 Base `  E )   &    |-  H  =  (  Hom  `  D )   &    |-  J  =  (  Hom  `  E )   &    |-  .1.  =  ( Id `  D )   &    |-  I  =  ( Id `  E )   &    |-  .x.  =  (comp `  D )   &    |-  O  =  (comp `  E )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   &    |-  ( ph  ->  F : B --> C )   &    |-  ( ph  ->  G  Fn  ( B  X.  B ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x G y ) : ( x H y ) --> ( ( F `
  x ) J ( F `  y
 ) ) )   &    |-  (
 ( ph  /\  x  e.  B )  ->  (
 ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  /\  ( m  e.  ( x H y )  /\  n  e.  ( y H z ) ) )  ->  ( ( x G z ) `  ( n ( <. x ,  y >.  .x.  z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) ) )   =>    |-  ( ph  ->  F ( D  Func  E ) G )
 
Theoremfuncf1 13584 The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  C  =  (
 Base `  E )   &    |-  ( ph  ->  F ( D 
 Func  E ) G )   =>    |-  ( ph  ->  F : B
 --> C )
 
Theoremfuncixp 13585* The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  H  =  ( 
 Hom  `  D )   &    |-  J  =  (  Hom  `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   =>    |-  ( ph  ->  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `  ( 1st `  z )
 ) J ( F `
  ( 2nd `  z
 ) ) )  ^m  ( H `  z ) ) )
 
Theoremfuncf2 13586 The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  H  =  ( 
 Hom  `  D )   &    |-  J  =  (  Hom  `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X G Y ) : ( X H Y )
 --> ( ( F `  X ) J ( F `  Y ) ) )
 
Theoremfuncfn2 13587 The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  ( ph  ->  F ( D  Func  E ) G )   =>    |-  ( ph  ->  G  Fn  ( B  X.  B ) )
 
Theoremfuncid 13588 A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  .1.  =  ( Id `  D )   &    |-  I  =  ( Id `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( ( X G X ) `  (  .1.  `  X ) )  =  ( I `  ( F `  X ) ) )
 
Theoremfuncco 13589 A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  H  =  ( 
 Hom  `  D )   &    |-  .x.  =  (comp `  D )   &    |-  O  =  (comp `  E )   &    |-  ( ph  ->  F ( D 
 Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  M  e.  ( X H Y ) )   &    |-  ( ph  ->  N  e.  ( Y H Z ) )   =>    |-  ( ph  ->  ( ( X G Z ) `  ( N (
 <. X ,  Y >.  .x. 
 Z ) M ) )  =  ( ( ( Y G Z ) `  N ) (
 <. ( F `  X ) ,  ( F `  Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) )
 
Theoremfuncsect 13590 The image of a section under a functor is a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  S  =  (Sect `  D )   &    |-  T  =  (Sect `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M ( X S Y ) N )   =>    |-  ( ph  ->  (
 ( X G Y ) `  M ) ( ( F `  X ) T ( F `  Y ) ) ( ( Y G X ) `  N ) )
 
Theoremfuncinv 13591 The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  I  =  (Inv `  D )   &    |-  J  =  (Inv `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M ( X I Y ) N )   =>    |-  ( ph  ->  (
 ( X G Y ) `  M ) ( ( F `  X ) J ( F `  Y ) ) ( ( Y G X ) `  N ) )
 
Theoremfunciso 13592 The image of an isomorphism under a functor is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  I  =  ( 
 Iso  `  D )   &    |-  J  =  (  Iso  `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M  e.  ( X I Y ) )   =>    |-  ( ph  ->  (
 ( X G Y ) `  M )  e.  ( ( F `  X ) J ( F `  Y ) ) )
 
Theoremfuncoppc 13593 A functor on categories yields a functor on the opposite categories (in the same direction). (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  P  =  (oppCat `  D )   &    |-  ( ph  ->  F ( C  Func  D ) G )   =>    |-  ( ph  ->  F ( O  Func  P )tpos 
 G )
 
Theoremidfuval 13594* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  H  =  (  Hom  `  C )   =>    |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `
  z ) ) ) >. )
 
Theoremidfu2nd 13595 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X ( 2nd `  I
 ) Y )  =  (  _I  |`  ( X H Y ) ) )
 
Theoremidfu2 13596 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  H  =  (  Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( ( X ( 2nd `  I
 ) Y ) `  F )  =  F )
 
Theoremidfu1st 13597 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   =>    |-  ( ph  ->  ( 1st `  I )  =  (  _I  |`  B ) )
 
Theoremidfu1 13598 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (
 ( 1st `  I ) `  X )  =  X )
 
Theoremidfucl 13599 The identity functor is a functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   =>    |-  ( C  e.  Cat  ->  I  e.  ( C  Func  C ) )
 
Theoremcofuval 13600* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   =>    |-  ( ph  ->  ( G  o.func 
 F )  =  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `  x ) ( 2nd `  G )
 ( ( 1st `  F ) `  y ) )  o.  ( x ( 2nd `  F )
 y ) ) )
 >. )
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