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Theorem List for Metamath Proof Explorer - 15001-15100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmre1cl 15001 In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  ->  X  e.  C )
 
Theoremmreintcl 15002 A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/= 
 (/) )  ->  |^| S  e.  C )
 
Theoremmreiincl 15003* A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  e.  C )
 
Theoremmrerintcl 15004 The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C )  ->  ( X  i^i  |^| S )  e.  C )
 
Theoremmreriincl 15005* The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C )
 
Theoremmreincl 15006 Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  ( A  i^i  B )  e.  C )
 
Theoremmreuni 15007 Since the entire base set of a Moore collection is the greatest element of it, the base set can be recovered from a Moore collection by set union. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  ->  U. C  =  X )
 
Theoremmreunirn 15008 Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  U. ran Moore  <->  C  e.  (Moore `  U. C ) )
 
Theoremismred 15009* Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( ph  ->  C  C_ 
 ~P X )   &    |-  ( ph  ->  X  e.  C )   &    |-  ( ( ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C )   =>    |-  ( ph  ->  C  e.  (Moore `  X )
 )
 
Theoremismred2 15010* Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ph  ->  C  C_ 
 ~P X )   &    |-  (
 ( ph  /\  s  C_  C )  ->  ( X  i^i  |^| s )  e.  C )   =>    |-  ( ph  ->  C  e.  (Moore `  X )
 )
 
Theoremmremre 15011 The Moore collections of subsets of a space, viewed as a kind of subset of the power set, form a Moore collection in their own right on the power set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( X  e.  V  ->  (Moore `  X )  e.  (Moore `  ~P X ) )
 
Theoremsubmre 15012 The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  ( C  i^i  ~P A )  e.  (Moore `  A ) )
 
7.2.1  Moore closures
 
Theoremmrcflem 15013* The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  ->  ( x  e.  ~P X  |->  |^|
 { s  e.  C  |  x  C_  s }
 ) : ~P X --> C )
 
Theoremfnmrc 15014 Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- mrCls  Fn  U. ran Moore
 
Theoremmrcfval 15015* Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
 
Theoremmrcf 15016 The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  F : ~P X --> C )
 
Theoremmrcval 15017* Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
 
Theoremmrccl 15018 The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  e.  C )
 
Theoremmrcsncl 15019 The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  e.  X )  ->  ( F `  { U } )  e.  C )
 
Theoremmrcid 15020 The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  ( F `  U )  =  U )
 
Theoremmrcssv 15021 The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  ( F `  U )  C_  X )
 
Theoremmrcidb 15022 A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  ( U  e.  C  <->  ( F `  U )  =  U ) )
 
Theoremmrcss 15023 Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  C_  ( F `  V ) )
 
Theoremmrcssid 15024 The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  ( F `  U ) )
 
Theoremmrcidb2 15025 A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( U  e.  C  <->  ( F `  U ) 
 C_  U ) )
 
Theoremmrcidm 15026 The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  ( F `
  U ) )  =  ( F `  U ) )
 
Theoremmrcsscl 15027 The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  U )  C_  V )
 
Theoremmrcuni 15028 Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  ~P X ) 
 ->  ( F `  U. U )  =  ( F ` 
 U. ( F " U ) ) )
 
Theoremmrcun 15029 Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  ( U  u.  V ) )  =  ( F `  (
 ( F `  U )  u.  ( F `  V ) ) ) )
 
Theoremmrcssvd 15030 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 15021. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   =>    |-  ( ph  ->  ( N `  B )  C_  X )
 
Theoremmrcssd 15031 Moore closure preserves subset ordering. Deduction form of mrcss 15023. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  V )   &    |-  ( ph  ->  V  C_  X )   =>    |-  ( ph  ->  ( N `  U )  C_  ( N `  V ) )
 
Theoremmrcssidd 15032 A set is contained in its Moore closure. Deduction form of mrcssid 15024. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  X )   =>    |-  ( ph  ->  U 
 C_  ( N `  U ) )
 
Theoremmrcidmd 15033 Moore closure is idempotent. Deduction form of mrcidm 15026. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  X )   =>    |-  ( ph  ->  ( N `  ( N `
  U ) )  =  ( N `  U ) )
 
Theoremmressmrcd 15034 In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  ( N `  T ) )   &    |-  ( ph  ->  T  C_  S )   =>    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )
 
Theoremsubmrc 15035 In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  F  =  (mrCls `  C )   &    |-  G  =  (mrCls `  ( C  i^i  ~P D ) )   =>    |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  =  ( F `  U ) )
 
Theoremmrieqvlemd 15036 In a Moore system, if  Y is a member of  S,  ( S 
\  { Y }
) and  S have the same closure if and only if  Y is in the closure of  ( S  \  { Y } ). Used in the proof of mrieqvd 15045 and mrieqv2d 15046. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  ( Y  e.  ( N `  ( S  \  { Y } ) )  <->  ( N `  ( S  \  { Y } ) )  =  ( N `  S ) ) )
 
7.2.2  Independent sets in a Moore system
 
Theoremmrisval 15037* Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   =>    |-  ( A  e.  (Moore `  X )  ->  I  =  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  (
 s  \  { x } ) ) }
 )
 
Theoremismri 15038* Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   =>    |-  ( A  e.  (Moore `  X )  ->  ( S  e.  I  <->  ( S  C_  X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
 ) ) ) ) )
 
Theoremismri2 15039* Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   =>    |-  ( ( A  e.  (Moore `  X )  /\  S  C_  X )  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) ) )
 
Theoremismri2d 15040* Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) ) )
 
Theoremismri2dd 15041* Definition of independence of a subset of the base set in a Moore system. One-way deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )   =>    |-  ( ph  ->  S  e.  I )
 
Theoremmriss 15042 An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.)
 |-  I  =  (mrInd `  A )   =>    |-  ( ( A  e.  (Moore `  X )  /\  S  e.  I )  ->  S  C_  X )
 
Theoremmrissd 15043 An independent set of a Moore system is a subset of the base set. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S  e.  I )   =>    |-  ( ph  ->  S 
 C_  X )
 
Theoremismri2dad 15044 Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  -.  Y  e.  ( N `  ( S  \  { Y }
 ) ) )
 
Theoremmrieqvd 15045* In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  ( N `  ( S 
 \  { x }
 ) )  =/=  ( N `  S ) ) )
 
Theoremmrieqv2d 15046* In a Moore system, a set is independent if and only if all its proper subsets have closure properly contained in the closure of the set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  A. s ( s  C.  S  ->  ( N `  s )  C.  ( N `  S ) ) ) )
 
Theoremmrissmrcd 15047 In a Moore system, if an independent set is between a set and its closure, the two sets are equal (since the two sets must have equal closures by mressmrcd 15034, and so are equal by mrieqv2d 15046.) (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  ( N `  T ) )   &    |-  ( ph  ->  T  C_  S )   &    |-  ( ph  ->  S  e.  I )   =>    |-  ( ph  ->  S  =  T )
 
Theoremmrissmrid 15048 In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  S )   =>    |-  ( ph  ->  T  e.  I )
 
Theoremmreexd 15049* In a Moore system, the closure operator is said to have the exchange property if, for all elements  y and  z of the base set and subsets  S of the base set such that  z is in the closure of  ( S  u.  { y } ) but not in the closure of  S,  y is in the closure of  ( S  u.  { z } ) (Definition 3.1.9 in [FaureFrolicher] p. 57 to 58.) This theorem allows us to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  Z  e.  ( N `  ( S  u.  { Y }
 ) ) )   &    |-  ( ph  ->  -.  Z  e.  ( N `  S ) )   =>    |-  ( ph  ->  Y  e.  ( N `  ( S  u.  { Z }
 ) ) )
 
Theoremmreexmrid 15050* In a Moore system whose closure operator has the exchange property, if a set is independent and an element is not in its closure, then adding the element to the set gives another independent set. Lemma 4.1.5 in [FaureFrolicher] p. 84. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  -.  Y  e.  ( N `  S ) )   =>    |-  ( ph  ->  ( S  u.  { Y }
 )  e.  I )
 
Theoremmreexexlemd 15051* This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 15055. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  X  e.  J )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  ( F  ~~  K  \/  G  ~~  K ) )   &    |-  ( ph  ->  A. t A. u  e.  ~P  ( X  \  t ) A. v  e.  ~P  ( X  \  t ) ( ( ( u  ~~  K  \/  v  ~~  K ) 
 /\  u  C_  ( N `  ( v  u.  t ) )  /\  ( u  u.  t
 )  e.  I ) 
 ->  E. i  e.  ~P  v ( u  ~~  i  /\  ( i  u.  t )  e.  I
 ) ) )   =>    |-  ( ph  ->  E. j  e.  ~P  G ( F  ~~  j  /\  ( j  u.  H )  e.  I )
 )
 
Theoremmreexexlem2d 15052* Used in mreexexlem4d 15054 to prove the induction step in mreexexd 15055. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  Y  e.  F )   =>    |-  ( ph  ->  E. g  e.  G  ( -.  g  e.  ( F  \  { Y } )  /\  (
 ( F  \  { Y } )  u.  ( H  u.  { g }
 ) )  e.  I
 ) )
 
Theoremmreexexlem3d 15053* Base case of the induction in mreexexd 15055. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  ( F  =  (/)  \/  G  =  (/) ) )   =>    |-  ( ph  ->  E. i  e.  ~P  G ( F  ~~  i  /\  ( i  u.  H )  e.  I )
 )
 
Theoremmreexexlem4d 15054* Induction step of the induction in mreexexd 15055. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  L  e.  om )   &    |-  ( ph  ->  A. h A. f  e. 
 ~P  ( X  \  h ) A. g  e.  ~P  ( X  \  h ) ( ( ( f  ~~  L  \/  g  ~~  L ) 
 /\  f  C_  ( N `  ( g  u.  h ) )  /\  ( f  u.  h )  e.  I )  ->  E. j  e.  ~P  g ( f  ~~  j  /\  ( j  u.  h )  e.  I
 ) ) )   &    |-  ( ph  ->  ( F  ~~  suc 
 L  \/  G  ~~  suc 
 L ) )   =>    |-  ( ph  ->  E. j  e.  ~P  G ( F  ~~  j  /\  ( j  u.  H )  e.  I )
 )
 
Theoremmreexexd 15055* Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if  F and  G are disjoint from  H,  ( F  u.  H ) is independent,  F is contained in the closure of  ( G  u.  H ), and either  F or  G is finite, then there is a subset  q of  G equinumerous to  F such that  ( q  u.  H ) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either  ( A  \  B ) or  ( B  \  A ) is finite. The theorem is proven by induction using mreexexlem3d 15053 for the base case and mreexexlem4d 15054 for the induction step. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  ( F  e.  Fin  \/  G  e.  Fin ) )   =>    |-  ( ph  ->  E. q  e.  ~P  G ( F  ~~  q  /\  ( q  u.  H )  e.  I )
 )
 
Theoremmreexdomd 15056* In a Moore system whose closure operator has the exchange property, if  S is independent and contained in the closure of  T, and either  S or  T is finite, then  T dominates  S. This is an immediate consequence of mreexexd 15055. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S 
 C_  ( N `  T ) )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( S  e.  Fin  \/  T  e.  Fin ) )   &    |-  ( ph  ->  S  e.  I
 )   =>    |-  ( ph  ->  S  ~<_  T )
 
Theoremmreexfidimd 15057* In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd 15056 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  e.  I
 )   &    |-  ( ph  ->  S  e.  Fin )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   =>    |-  ( ph  ->  S 
 ~~  T )
 
7.2.3  Algebraic closure systems
 
Theoremisacs 15058* A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  E. f ( f : ~P X --> ~P X  /\  A. s  e.  ~P  X ( s  e.  C  <->  U. ( f "
 ( ~P s  i^i 
 Fin ) )  C_  s ) ) ) )
 
Theoremacsmre 15059 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  ->  C  e.  (Moore `  X )
 )
 
Theoremisacs2 15060* In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( s  e.  C  <->  A. y  e.  ( ~P s  i^i  Fin )
 ( F `  y
 )  C_  s )
 ) )
 
Theoremacsfiel 15061* 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 15062* A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  ( S  e.  C  <->  A. y  e.  ( ~P S  i^i  Fin )
 ( F `  y
 )  C_  S )
 )
 
Theoremacsmred 15063 An algebraic closure system is also a Moore system. Deduction form of acsmre 15059. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   =>    |-  ( ph  ->  A  e.  (Moore `  X )
 )
 
Theoremisacs1i 15064* 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 15065 Algebraicity is a composable 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 15066* 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 15067* 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 15068* 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 15069* 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 15070* 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 15071 Extend class notation with the class of categories.
 class  Cat
 
Syntaxccid 15072 Extend class notation with the identity arrow of a category.
 class  Id
 
Syntaxchomf 15073 Extend class notation to include functionalized Hom-set extractor.
 class  Hom f
 
Syntaxccomf 15074 Extend class notation to include functionalized composition operation.
 class compf
 
Definitiondf-cat 15075* 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. Definition in [Lang] p. 53. In contrast to definition 3.1 of [Adamek] p. 21, where "A category is a quadruple A = (O, hom, id, o)", a category is defined as an extensible structure consisting of three slots: the objects "O" ( ( Base `  c
)), the morphisms "hom" ( ( Hom  `  c )) and the composition law "o" (
(comp `  c )). The identities "id" are defined by their properties related to morphisms and their composition, see condition 3.1(b) in [Adamek] p. 21 and df-cid 15076. (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 15076* 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 15077* 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.)
 |- 
 Hom f  =  ( c  e.  _V  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c )  |->  ( x ( Hom  `  c
 ) y ) ) )
 
Definitiondf-comf 15078* 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 15079* 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 15080* 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 15081* 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 15082* 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 15083* 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 15084* 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 15085 The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |- 
 Id  Fn  Cat
 
Theoremcidfn 15086 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 15087* 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 15088* Version of iscatd 15080 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 15089 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 15090 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 15091 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 15092 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 15093 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 15094 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 15095 The empty set is a category, the empty category, see example 3.3(4.c) in [Adamek] p. 24. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  (/)  e.  Cat
 
Theoremhomffval 15096* Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  F  =  ( Hom f  `  C )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   =>    |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )
 
Theoremfnhomeqhomf 15097 If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.)
 |-  F  =  ( Hom f  `  C )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( H  Fn  ( B  X.  B )  ->  F  =  H )
 
Theoremhomfval 15098 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  F  =  ( Hom f  `  C )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X F Y )  =  ( X H Y ) )
 
Theoremhomffn 15099 The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  F  =  ( Hom f  `  C )   &    |-  B  =  (
 Base `  C )   =>    |-  F  Fn  ( B  X.  B )
 
Theoremhomfeq 15100* 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  ->  (
 ( Hom f  `  C )  =  ( Hom f  `  D )  <->  A. x  e.  B  A. y  e.  B  ( x H y )  =  ( x J y ) ) )
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