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Theorem List for Metamath Proof Explorer - 13501-13600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubmre 13501 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 13502* 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 13503 Moore-closure is a well behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- mrCls  Fn  U. ran Moore
 
Theoremmrcfval 13504* 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 13505 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 13506* 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 13507 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 13508 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 13509 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 13510 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 13511 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 13512 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 13513 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 13514 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 13515 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 13516 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 13517 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 13518 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 13519 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 13510. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   =>    |-  ( ph  ->  ( N `  B )  C_  X )
 
Theoremmrcssd 13520 Moore closure preserves subset ordering. Deduction form of mrcss 13512. (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 13521 A set is contained in its Moore closure. Deduction form of mrcssid 13513. (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 13522 Moore closure is idempotent. Deduction form of mrcidm 13515. (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 13523 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 13524 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 13525 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 13534 and mrieqv2d 13535. 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 13526* 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 13527* 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 13528* 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 13529* 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 13530* 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 13531 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 13532 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 13533 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 13534* 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 13535* 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 13536 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 13523, and so are equal by mrieqv2d 13535.) (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 13537 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 13538* 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 13539* 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 13540* This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 13544. (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 13541* Used in mreexexlem4d 13543 to prove the induction step in mreexexd 13544. 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 13542* Base case of the induction in mreexexd 13544. (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 13543* Induction step of the induction in mreexexd 13544. (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 13544* 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 13542 for the base case and mreexexlem4d 13543 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 13545* 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 13544. (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 13546* 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 13545 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 13547* 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 13548 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  ->  C  e.  (Moore `  X )
 )
 
Theoremisacs2 13549* 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 13550* 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 13551* 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 13552 An algebraic closure system is also a Moore system. Deduction form of acsmre 13548. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   =>    |-  ( ph  ->  A  e.  (Moore `  X )
 )
 
Theoremisacs1i 13553* 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 13554 Algebraicity is a composible property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  e.  V  ->  (ACS `  X )  e.  (Moore `  ~P X ) )
 
Theoremacsfn 13555* 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 13556* 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 13557* 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 13558* 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 13559* 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 13560 Extend class notation with the class of categories.
 class  Cat
 
Syntaxccid 13561 Extend class notation with the identity arrow of a category.
 class  Id
 
Syntaxchomf 13562 Extend class notation to include functionalized Hom-set extractor.
 class  Homf
 
Syntaxccomf 13563 Extend class notation to include functionalized composition operation.
 class compf
 
Definitiondf-cat 13564* A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated to those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
 |- 
 Cat  =  { c  |  [. ( Base `  c
 )  /  b ]. [. (  Hom  `  c
 )  /  h ]. [. (comp `  c )  /  o ]. A. x  e.  b  ( E. g  e.  ( x h x ) A. y  e.  b  ( A. f  e.  (
 y h x ) ( g ( <. y ,  x >. o x ) f )  =  f  /\  A. f  e.  ( x h y ) ( f (
 <. x ,  x >. o y ) g )  =  f )  /\  A. y  e.  b  A. z  e.  b  A. f  e.  ( x h y ) A. g  e.  ( y h z ) ( ( g ( <. x ,  y >. o z ) f )  e.  ( x h z )  /\  A. w  e.  b  A. k  e.  ( z h w ) ( ( k ( <. y ,  z >. o w ) g ) ( <. x ,  y >. o w ) f )  =  ( k ( <. x ,  z >. o w ) ( g ( <. x ,  y >. o z ) f ) ) ) ) }
 
Definitiondf-cid 13565* 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 13566* Define the functionalized Hom-set operator, which is exactly like  Hom but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |- 
 Homf  =  ( c  e.  _V  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c )  |->  ( x (  Hom  `  c
 ) y ) ) )
 
Definitiondf-comf 13567* 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 13568* 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 13569* 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 13570* 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 13571* 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 13572* 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 13573* 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 13574 The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |- 
 Id  Fn  Cat
 
Theoremcidfn 13575 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 13576* 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 13577* Version of iscatd2 13577 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 13578 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 13579 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 13580 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 13581 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 13582 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 13583 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 13584 The empty set is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  (/)  e.  Cat
 
Theoremproplem2 13585* Lemma for mndpropd 14392. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  ( ( ( X  e.  A  /\  Y  e.  B )  /\  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C )  ->  ( X F Y )  e.  C )
 
Theoremproplem 13586* Lemma for mndpropd 14392. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ( x F y )  =  ( x G y ) )   =>    |-  ( ( ph  /\  ( X  e.  A  /\  Y  e.  B )
 )  ->  ( X F Y )  =  ( X G Y ) )
 
Theoremproplem3 13587 Lemma for property theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ( ph  /\  ps )  ->  ( x F y )  =  ( x G y ) )
 
Theoremhomffval 13588* Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  F  =  (  Homf  `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   =>    |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )
 
Theoremhomfval 13589 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  F  =  (  Homf  `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X F Y )  =  ( X H Y ) )
 
Theoremhomffn 13590 The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  F  =  (  Homf  `  C )   &    |-  B  =  ( Base `  C )   =>    |-  F  Fn  ( B  X.  B )
 
Theoremhomfeq 13591* Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  H  =  (  Hom  `  C )   &    |-  J  =  ( 
 Hom  `  D )   &    |-  ( ph  ->  B  =  (
 Base `  C ) )   &    |-  ( ph  ->  B  =  ( Base `  D )
 )   =>    |-  ( ph  ->  (
 (  Homf  `  C )  =  ( 
 Homf  `  D )  <->  A. x  e.  B  A. y  e.  B  ( x H y )  =  ( x J y ) ) )
 
Theoremhomfeqd 13592 If two structures have the same 
Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  ( Base `  C )  =  ( Base `  D )
 )   &    |-  ( ph  ->  (  Hom  `  C )  =  (  Hom  `  D ) )   =>    |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )
 
Theoremhomfeqbas 13593 Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )   =>    |-  ( ph  ->  (
 Base `  C )  =  ( Base `  D )
 )
 
Theoremhomfeqval 13594 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  ( ph  ->  (  Homf  `  C )  =  ( 
 Homf  `  D ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X H Y )  =  ( X J Y ) )
 
Theoremcomfffval 13595* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   =>    |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `
  x )  |->  ( g ( x  .x.  y ) f ) ) )
 
Theoremcomffval 13596* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y )  |->  ( g ( <. X ,  Y >.  .x.  Z )
 f ) ) )
 
Theoremcomfval 13597 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Hom  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  ( G ( <. X ,  Y >.  .x.  Z ) F ) )
 
Theoremcomfffval2 13598* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Homf  `  C )   &    |-  .x.  =  (comp `  C )   =>    |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `
  x )  |->  ( g ( x  .x.  y ) f ) ) )
 
Theoremcomffval2 13599* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (compf `  C )   &    |-  B  =  ( Base `  C )   &    |-  H  =  ( 
 Homf  `  C )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y )  |->  ( g ( <. X ,  Y >.  .x.  Z )
 f ) ) )
 
Theoremcomfval2 13600 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 ) )
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