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Theorem List for Metamath Proof Explorer - 25601-25700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisfne4 25601 The predicate " B is finer than  A " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B ) ) )
 
Theoremisfne4b 25602 A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  V  ->  ( A Fne B  <->  ( X  =  Y  /\  ( topGen `  A )  C_  ( topGen `  B )
 ) ) )
 
Theoremisfne2 25603* The predicate " B is finer than  A." (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  A. y  e.  x  E. z  e.  B  (
 y  e.  z  /\  z  C_  x ) ) ) )
 
Theoremisfne3 25604* The predicate " B is finer than  A." (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  E. y ( y  C_  B  /\  x  =  U. y ) ) ) )
 
Theoremfnebas 25605 A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( A Fne B  ->  X  =  Y )
 
Theoremfnetg 25606 A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( A Fne B  ->  A  C_  ( topGen `  B )
 )
 
Theoremfnessex 25607* If  B is finer than  A and  S is an element of  A, every point in  S is an element of a subset of  S which is in  B. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  (
 ( A Fne B  /\  S  e.  A  /\  P  e.  S )  ->  E. x  e.  B  ( P  e.  x  /\  x  C_  S ) )
 
Theoremfneuni 25608* If  B is finer than  A, every element of  A is a union of elements of  B. (Contributed by Jeff Hankins, 11-Oct-2009.)
 |-  (
 ( A Fne B  /\  S  e.  A ) 
 ->  E. x ( x 
 C_  B  /\  S  =  U. x ) )
 
Theoremfneint 25609* If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)
 |-  ( A Fne B  ->  |^| { x  e.  B  |  P  e.  x }  C_  |^| { x  e.  A  |  P  e.  x } )
 
Theoremrefrel 25610 Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  Rel  Ref
 
Theoremisref 25611* The property of being a refinement of a cover. Dr. Nyikos once commented in class that the term "refinement" is actually misleading and that people are inclined to confuse it with the notion defined in isfne 25600. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Ref B  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) ) )
 
Theoremrefbas 25612 A refinement covers the same set. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( A Ref B  ->  X  =  Y )
 
Theoremrefssex 25613* Every set in a refinement has a superset in the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  (
 ( A Ref B  /\  S  e.  B ) 
 ->  E. x  e.  A  S  C_  x )
 
Theoremfness 25614 A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( ( B  e.  C  /\  A  C_  B  /\  X  =  Y ) 
 ->  A Fne B )
 
Theoremssref 25615 A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y ) 
 ->  B Ref A )
 
Theoremfneref 25616 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
 |-  ( A  e.  V  ->  A Fne A )
 
Theoremrefref 25617 Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  ( A  e.  V  ->  A Ref A )
 
Theoremfnetr 25618 Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  (
 ( A Fne B  /\  B Fne C ) 
 ->  A Fne C )
 
Theoremfneval 25619 Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  .~  B 
 <->  ( topGen `  A )  =  ( topGen `  B )
 ) )
 
Theoremfneer 25620 Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |- 
 .~  Er  _V
 
Theoremreftr 25621 Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  (
 ( A Ref B  /\  B Ref C ) 
 ->  A Ref C )
 
Theoremtopfne 25622 Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( K  e.  Top  /\  X  =  Y ) 
 ->  ( J  C_  K  <->  J Fne K ) )
 
Theoremtopfneec 25623 A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |-  ( J  e.  Top  ->  ( A  e.  [ J ]  .~  <->  ( topGen `  A )  =  J )
 )
 
Theoremtopfneec2 25624 A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( [ J ]  .~  =  [ K ]  .~  <->  J  =  K ) )
 
Theoremfnessref 25625* A cover is finer iff it has a subcover which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( X  =  Y  ->  ( A Fne B  <->  E. c ( c  C_  B  /\  A ( Fne 
 i^i  Ref ) c ) ) )
 
Theoremrefssfne 25626* A cover is a refinement iff it is a subcover of something which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( X  =  Y  ->  ( A Ref B  <->  E. c ( B  C_  c  /\  A ( Fne 
 i^i  Ref ) c ) ) )
 
Theoremisptfin 25627* The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. A   =>    |-  ( A  e.  B  ->  ( A  e.  PtFin  <->  A. x  e.  X  { y  e.  A  |  x  e.  y }  e.  Fin ) )
 
Theoremislocfin 25628* The statement "is a locally finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. J   &    |-  Y  =  U. A   =>    |-  ( A  e.  ( LocFin `
  J )  <->  ( J  e.  Top  /\  X  =  Y  /\  A. x  e.  X  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
 
Theoremfinptfin 25629 A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  ( A  e.  Fin  ->  A  e.  PtFin )
 
Theoremptfinfin 25630* A point covered by a point-finite cover is only covered by finitely many elements. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. A   =>    |-  ( ( A  e.  PtFin  /\  P  e.  X ) 
 ->  { x  e.  A  |  P  e.  x }  e.  Fin )
 
Theoremfinlocfin 25631 A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. J   &    |-  Y  =  U. A   =>    |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  A  e.  ( LocFin `  J ) )
 
Theoremlocfintop 25632 A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  ( A  e.  ( LocFin `  J )  ->  J  e.  Top )
 
Theoremlocfinbas 25633 A locally finite cover must cover the base set of its corresponding topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. J   &    |-  Y  =  U. A   =>    |-  ( A  e.  ( LocFin `
  J )  ->  X  =  Y )
 
Theoremlocfinnei 25634* A point covered by a locally finite cover has a neighborhood which intersects only finitely many elements of the cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. J   =>    |-  ( ( A  e.  ( LocFin `  J )  /\  P  e.  X ) 
 ->  E. n  e.  J  ( P  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
 
Theoremlfinpfin 25635 A locally finite cover is point-finite. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  ( A  e.  ( LocFin `  J )  ->  A  e.  PtFin
 )
 
Theoremlocfincmp 25636 For a compact space, the locally finite covers are precisely the finite covers. Sadly, this property does not properly characterize all compact spaces. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. C   =>    |-  ( J  e.  Comp  ->  ( C  e.  ( LocFin `
  J )  <->  ( C  e.  Fin  /\  X  =  Y ) ) )
 
Theoremlocfindis 25637 The locally finite covers of a discrete space are precisely the point-finite covers. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  Y  =  U. C   =>    |-  ( C  e.  ( LocFin `
  ~P X )  <-> 
 ( C  e.  PtFin  /\  X  =  Y ) )
 
Theoremlocfincf 25638 A locally finite cover in a coarser topology is locally finite in a finer topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. J   =>    |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( LocFin `  J )  C_  ( LocFin `  K )
 )
 
Theoremcomppfsc 25639* A space where every open cover has a point-finite subcover is compact. This is significant in part because it shows half of the proposition that if only half the generalization in the definition of metacompactness (and consequently paracompactness) is performed, one does not obtain any more spaces. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( J  e.  Comp  <->  A. c  e.  ~P  J ( X  =  U. c  ->  E. d  e.  PtFin  ( d  C_  c  /\  X  =  U. d ) ) ) )
 
16.13.5  Neighborhood bases determine topologies
 
Theoremneibastop1 25640* A collection of neighborhood bases determines a topology. Part of Theorem 4.5 of Stephen Willard's General Topology. (Contributed by Jeff Hankins, 8-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( ~P ~P X  \  { (/) } )
 )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x )  /\  w  e.  ( F `  x ) ) )  ->  ( ( F `  x )  i^i  ~P (
 v  i^i  w )
 )  =/=  (/) )   &    |-  J  =  { o  e.  ~P X  |  A. x  e.  o  ( ( F `
  x )  i^i 
 ~P o )  =/=  (/) }   =>    |-  ( ph  ->  J  e.  (TopOn `  X )
 )
 
Theoremneibastop2lem 25641* Lemma for neibastop2 25642. (Contributed by Jeff Hankins, 12-Sep-2009.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( ~P ~P X  \  { (/) } )
 )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x )  /\  w  e.  ( F `  x ) ) )  ->  ( ( F `  x )  i^i  ~P (
 v  i^i  w )
 )  =/=  (/) )   &    |-  J  =  { o  e.  ~P X  |  A. x  e.  o  ( ( F `
  x )  i^i 
 ~P o )  =/=  (/) }   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  x  e.  v )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  E. t  e.  ( F `  x ) A. y  e.  t  ( ( F `  y )  i^i  ~P v
 )  =/=  (/) )   &    |-  ( ph  ->  P  e.  X )   &    |-  ( ph  ->  N  C_  X )   &    |-  ( ph  ->  U  e.  ( F `  P ) )   &    |-  ( ph  ->  U  C_  N )   &    |-  G  =  ( rec ( ( a  e. 
 _V  |->  U_ z  e.  a  U_ x  e.  X  ( ( F `  x )  i^i  ~P z ) ) ,  { U } )  |`  om )   &    |-  S  =  { y  e.  X  |  E. f  e.  U. ran  G ( ( F `
  y )  i^i 
 ~P f )  =/=  (/) }   =>    |-  ( ph  ->  E. u  e.  J  ( P  e.  u  /\  u  C_  N ) )
 
Theoremneibastop2 25642* In the topology generated by a neighborhood base, a set is a neighborhood of a point iff it contains a subset in the base. (Contributed by Jeff Hankins, 9-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( ~P ~P X  \  { (/) } )
 )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x )  /\  w  e.  ( F `  x ) ) )  ->  ( ( F `  x )  i^i  ~P (
 v  i^i  w )
 )  =/=  (/) )   &    |-  J  =  { o  e.  ~P X  |  A. x  e.  o  ( ( F `
  x )  i^i 
 ~P o )  =/=  (/) }   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  x  e.  v )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  E. t  e.  ( F `  x ) A. y  e.  t  ( ( F `  y )  i^i  ~P v
 )  =/=  (/) )   =>    |-  ( ( ph  /\  P  e.  X ) 
 ->  ( N  e.  (
 ( nei `  J ) `  { P } )  <->  ( N  C_  X  /\  ( ( F `  P )  i^i  ~P N )  =/=  (/) ) ) )
 
Theoremneibastop3 25643* The topology generated by a neighborhood base is unique. (Contributed by Jeff Hankins, 16-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( ~P ~P X  \  { (/) } )
 )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x )  /\  w  e.  ( F `  x ) ) )  ->  ( ( F `  x )  i^i  ~P (
 v  i^i  w )
 )  =/=  (/) )   &    |-  J  =  { o  e.  ~P X  |  A. x  e.  o  ( ( F `
  x )  i^i 
 ~P o )  =/=  (/) }   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  x  e.  v )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  E. t  e.  ( F `  x ) A. y  e.  t  ( ( F `  y )  i^i  ~P v
 )  =/=  (/) )   =>    |-  ( ph  ->  E! j  e.  (TopOn `  X ) A. x  e.  X  ( ( nei `  j ) `  { x } )  =  { n  e.  ~P X  |  ( ( F `  x )  i^i  ~P n )  =/=  (/) } )
 
16.13.6  Lattice structure of topologies
 
Theoremtopmtcl 25644 The meet of a collection of topologies on  X is again a topology on  X. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  S  C_  (TopOn `  X ) )  ->  ( ~P X  i^i  |^| S )  e.  (TopOn `  X ) )
 
Theoremtopmeet 25645* Two equivalent formulations of the meet of a collection of topologies. (Contributed by Jeff Hankins, 4-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  S  C_  (TopOn `  X ) )  ->  ( ~P X  i^i  |^| S )  =  U. { k  e.  (TopOn `  X )  |  A. j  e.  S  k  C_  j } )
 
Theoremtopjoin 25646* Two equivalent formulations of the join of a collection of topologies. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  S  C_  (TopOn `  X ) )  ->  ( topGen `  ( fi `  ( { X }  u.  U. S ) ) )  = 
 |^| { k  e.  (TopOn `  X )  |  A. j  e.  S  j  C_  k } )
 
Theoremfnemeet1 25647* The meet of a collection of equivalence classes of covers with respect to fineness. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  A. y  e.  S  X  =  U. y  /\  A  e.  S )  ->  ( ~P X  i^i  |^|_
 t  e.  S  (
 topGen `  t ) ) Fne A )
 
Theoremfnemeet2 25648* The meet of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  A. y  e.  S  X  =  U. y ) 
 ->  ( T Fne ( ~P X  i^i  |^|_ t  e.  S  ( topGen `  t
 ) )  <->  ( X  =  U. T  /\  A. x  e.  S  T Fne x ) ) )
 
Theoremfnejoin1 25649* Join of equivalence classes under the fineness relation-part one. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  A. y  e.  S  X  =  U. y  /\  A  e.  S )  ->  A Fne if ( S  =  (/) ,  { X } ,  U. S ) )
 
Theoremfnejoin2 25650* Join of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  A. y  e.  S  X  =  U. y ) 
 ->  ( if ( S  =  (/) ,  { X } ,  U. S ) Fne T  <->  ( X  =  U. T  /\  A. x  e.  S  x Fne T ) ) )
 
16.13.7  Filter bases
 
Theoremfgmin 25651 Minimality property of a generated filter: every filter that contains  B contains its generated filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
 |-  (
 ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X ) )  ->  ( B  C_  F  <->  ( X filGen B )  C_  F )
 )
 
Theoremneifg 25652* The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 17464. (Contributed by Jeff Hankins, 3-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( X filGen { x  e.  J  |  S  C_  x } )  =  ( ( nei `  J ) `  S ) )
 
16.13.8  Directed sets, nets
 
Theoremtailfval 25653* The tail function for a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  X  =  dom  D   =>    |-  ( D  e.  DirRel  ->  ( tail `  D )  =  ( x  e.  X  |->  ( D " { x } ) ) )
 
Theoremtailval 25654 The tail of an element in a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  X  =  dom  D   =>    |-  ( ( D  e.  DirRel  /\  A  e.  X ) 
 ->  ( ( tail `  D ) `  A )  =  ( D " { A } ) )
 
Theoremeltail 25655 An element of a tail. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  X  =  dom  D   =>    |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  (
 ( tail `  D ) `  A )  <->  A D B ) )
 
Theoremtailf 25656 The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  X  =  dom  D   =>    |-  ( D  e.  DirRel  ->  ( tail `  D ) : X --> ~P X )
 
Theoremtailini 25657 A tail contains its initial element. (Contributed by Jeff Hankins, 25-Nov-2009.)
 |-  X  =  dom  D   =>    |-  ( ( D  e.  DirRel  /\  A  e.  X ) 
 ->  A  e.  ( (
 tail `  D ) `  A ) )
 
Theoremtailfb 25658 The collection of tails of a directed set is a filter base. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  X  =  dom  D   =>    |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ran  ( tail `  D )  e.  ( fBas `  X )
 )
 
Theoremfilnetlem1 25659* Lemma for filnet 25663. Change variables. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  H  =  U_ n  e.  F  ( { n }  X.  n )   &    |-  D  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  ( 1st `  y )  C_  ( 1st `  x )
 ) }   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A D B  <->  ( ( A  e.  H  /\  B  e.  H ) 
 /\  ( 1st `  B )  C_  ( 1st `  A ) ) )
 
Theoremfilnetlem2 25660* Lemma for filnet 25663. The field of the direction. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  H  =  U_ n  e.  F  ( { n }  X.  n )   &    |-  D  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  ( 1st `  y )  C_  ( 1st `  x )
 ) }   =>    |-  ( (  _I  |`  H ) 
 C_  D  /\  D  C_  ( H  X.  H ) )
 
Theoremfilnetlem3 25661* Lemma for filnet 25663. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  H  =  U_ n  e.  F  ( { n }  X.  n )   &    |-  D  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  ( 1st `  y )  C_  ( 1st `  x )
 ) }   =>    |-  ( H  =  U. U. D  /\  ( F  e.  ( Fil `  X )  ->  ( H  C_  ( F  X.  X ) 
 /\  D  e.  DirRel ) ) )
 
Theoremfilnetlem4 25662* Lemma for filnet 25663. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  H  =  U_ n  e.  F  ( { n }  X.  n )   &    |-  D  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  ( 1st `  y )  C_  ( 1st `  x )
 ) }   =>    |-  ( F  e.  ( Fil `  X )  ->  E. d  e.  DirRel  E. f
 ( f : dom  d
 --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  d ) ) ) )
 
Theoremfilnet 25663* A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  E. d  e.  DirRel  E. f ( f : dom  d --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  d )
 ) ) )
 
16.14  Mathbox for Jeff Madsen
 
16.14.1  Logic and set theory
 
Theoremanim12da 25664 Conjoin antecedents and consequents in a deduction. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( ph  /\  ps )  ->  th )   &    |-  ( ( ph  /\ 
 ch )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ch ) )  ->  ( th  /\  ta )
 )
 
Theorembiadan2OLD 25665 Add a conjunction to an equivalence. (Moved to biadan2 626 in main set.mm and may be deleted by mathbox owner, JM. --NM 25-Feb-2014.) (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ph  <->  ( ps  /\  ch ) )
 
Theoremsyldanl 25666 A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   &    |-  ( ( (
 ph  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ph  /\  ps )  /\  th )  ->  ta )
 
Theoremsyl3an2cOLDOLD 25667 A syllogism inference combined with contraction. (Moved to syl13anc 1189 in main set.mm and may be deleted by mathbox owner, JM. --NM 8-Sep-2011.) (Contributed by Jeff Madsen, 17-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ( ps 
 /\  ch  /\  th )
 )  ->  ta )   &    |-  ( et  ->  ph )   &    |-  ( et  ->  ps )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  th )   =>    |-  ( et  ->  ta )
 
Theoremmp4anOLD 25668 An inference based on modus ponens. (Moved to mp4an 657 in main set.mm and may be deleted by mathbox owner, JM. --NM 19-Oct-2012.) (Contributed by Jeff Madsen, 15-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ch   &    |-  th   &    |-  ( ( (
 ph  /\  ps )  /\  ( ch  /\  th ) )  ->  ta )   =>    |-  ta
 
TheoremimdistandaOLD 25669 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Moved into main set.mm as imdistanda 677 and may be deleted by mathbox owner, SF. --NM 20-Sep-2013.) (Contributed by Jeff Madsen, 19-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  ( ch  ->  th )
 )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  ( ps  /\  th ) ) )
 
Theoremr19.21aivvaOLD 25670* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Moved to ralrimivva 2606 in main set.mm and may be deleted by mathbox owner, JM. --NM 17-Jul-2012.) (Contributed by Jeff Madsen, 19-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  ps )   =>    |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )
 
TheoremsbmoOLD 25671* Substitution into "at most one". (Moved to sbmo 2146 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Jan-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
 
Theorem2ralorOLD 25672* Distribute quantification over "or". (Moved to 2ralor 2680 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Jan-2012.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  e.  A  A. y  e.  B  (
 ph  \/  ps )  <->  (
 A. x  e.  A  ph 
 \/  A. y  e.  B  ps ) )
 
TheoremrexunOLD 25673 Restricted existential quantification over union. (Moved to rexun 3297 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Jan-2012.) (Contributed by Jeff Madsen, 5-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x  e.  ( A  u.  B ) ph  <->  ( E. x  e.  A  ph 
 \/  E. x  e.  B  ph ) )
 
TheoremralunOLD 25674* Restricted quantification over union. (Moved to ralun 3299 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Jan-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A. x  e.  A  ph 
 /\  A. x  e.  B  ph )  ->  A. x  e.  ( A  u.  B ) ph )
 
TheoremrexsnOLD 25675* Restricted existential quantification over a singleton. (Moved to rexsn 3616 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Jan-2012.) (Contributed by Jeff Madsen, 5-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e. 
 { A } ph  <->  ps )
 
Theoremrexcom4aOLD 25676* Specialized existential commutation lemma. (Moved to rexcom4a 2759 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Apr-2012.) (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x E. y  e.  A  ( ph  /\  ps ) 
 <-> 
 E. y  e.  A  ( ph  /\  E. x ps ) )
 
Theoremrexcom4bOLD 25677* Specialized existential commutation lemma. (Moved to rexcom4b 2760 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Apr-2012.) (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  e.  _V   =>    |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B ) 
 <-> 
 E. y  e.  A  ph )
 
Theorem3reeanvOLD 25678* Rearrange three existential quantifiers. (Moved to 3reeanv 2679 in main set.mm and may be deleted by mathbox owner, JM. --NM 21-Mar-2013.) (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  ( ph  /\  ps  /\  ch )  <->  ( E. x  e.  A  ph  /\  E. y  e.  B  ps  /\  E. z  e.  C  ch ) )
 
TheoremmorexOLD 25679* Derive membership from uniqueness. (Moved to morex 2900 in main set.mm and may be deleted by mathbox owner, JM. --NM 19-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  e.  _V   &    |-  ( x  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( ( E. x  e.  A  ph  /\  E* x ph )  ->  ( ps  ->  B  e.  A ) )
 
Theoremeuuni2OLD 25680* The unique element such that 
ph. (Moved to iota2 6216 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( A  e.  B  /\  E! x ph )  ->  ( ps 
 <-> 
 U. { x  |  ph
 }  =  A ) )
 
Theoremunirep 25681* Define a quantity whose definition involves a choice of representative, but which is uniquely determined regardless of the choice. (Contributed by Jeff Madsen, 1-Jun-2011.)
 |-  (
 y  =  D  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  D  ->  B  =  C )   &    |-  (
 y  =  z  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  z  ->  B  =  F )   &    |-  B  e.  _V   =>    |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\ 
 ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  ( iota x E. y  e.  A  ( ph  /\  x  =  B ) )  =  C )
 
Theoremrabeq12OLD 25682* Equality of restricted class abstractions. (Moved to rabeqbidv 2735 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Sep-2011.) (Contributed by Jeff Madsen, 1-Dec-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch }
 )
 
Theoremrabeq0OLD 25683* Condition for a restricted class abstraction to be empty. (Moved to rabeq0 3418 in main set.mm and may be deleted by mathbox owner, JM. --NM 1-Apr-2013.) (Contributed by Jeff Madsen, 7-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( { x  e.  A  |  ph }  =  (/)  <->  A. x  e.  A  -.  ph )
 
TheoremrabxmOLD 25684* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) (Moved to rabxm 3419 in main set.mm and may be deleted by mathbox owner, JM. --NM 10-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  =  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  -.  ph } )
 
TheoremrabncOLD 25685* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) (Moved to rabnc 3420 in main set.mm and may be deleted by mathbox owner, JM. --NM 10-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  (/)
 
TheoremralabOLD 25686* Universal quantification over a class abstraction. (Moved to ralab 2877 in main set.mm and may be deleted by mathbox owner, JM. --NM 20-Oct-2011.) (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 y  =  x  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  { y  |  ph } ch  <->  A. x ( ps 
 ->  ch ) )
 
TheoremralrabOLD 25687* Universal quantification over a restricted class abstraction. (Moved to ralrab 2878 in main set.mm and may be deleted by mathbox owner, JM. --NM 20-Oct-2011.) (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 y  =  x  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  { y  e.  A  |  ph } ch  <->  A. x  e.  A  ( ps  ->  ch )
 )
 
TheoremrexrabOLD 25688* Existential quantification over a class abstraction. (Moved to rexrab 2880 in main set.mm and may be deleted by mathbox owner, JM. --NM 20-Mar-2013.) (Contributed by Jeff Madsen, 17-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. y  e.  { x  e.  A  |  ph } ch  <->  E. y  e.  A  ( ps  /\  ch )
 )
 
TheoremraleqfnOLD 25689* Change the domain of quantification by a function. (Moved to ralima 5657 in main set.mm and may be deleted by mathbox owner, JM. --NM 17-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 y  =  ( F `
  x )  ->  ( ph  <->  ps ) )   =>    |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( A. x  e.  B  ps 
 <-> 
 A. y  e.  ( F " B ) ph ) )
 
Theoremcover2 25690* Two ways of expressing the statement "there is a cover of  A by elements of  B such that for each set in the cover,  ph." Note that  ph and  x must be distinct. (Contributed by Jeff Madsen, 20-Jun-2010.)
 |-  B  e.  _V   &    |-  A  =  U. B   =>    |-  ( A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  ph )  <->  E. z  e.  ~P  B ( U. z  =  A  /\  A. y  e.  z  ph ) )
 
Theoremcover2g 25691* Two ways of expressing the statement "there is a cover of  A by elements of  B such that for each set in the cover,  ph." Note that  ph and  x must be distinct. (Contributed by Jeff Madsen, 21-Jun-2010.)
 |-  A  =  U. B   =>    |-  ( B  e.  C  ->  ( A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  ph )  <->  E. z  e.  ~P  B ( U. z  =  A  /\  A. y  e.  z  ph ) ) )
 
TheoremdisjrOLD 25692* Two ways of saying that two classes are disjoint. (Moved to disjr 3438 in main set.mm and may be deleted by mathbox owner, JM. --NM 23-Mar-2013.) (Contributed by Jeff Madsen, 19-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  i^i  B )  =  (/)  <->  A. x  e.  B  -.  x  e.  A )
 
Theoremdifin2OLD 25693 Represent a set difference as an intersection with a larger difference. (Moved to difin2 3372 in main set.mm and may be deleted by mathbox owner, JM. --NM 31-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  C  ->  ( A  \  B )  =  ( ( C  \  B )  i^i  A ) )
 
TheoremprfunOLD 25694 A function with a domain of two elements. (Moved to funpr 5205 in main set.mm and may be deleted by mathbox owner, JM. --NM 26-Aug-2011.) (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  Fun  { <. A ,  C >. ,  <. B ,  D >. } )
 
Theoremprfv1OLD 25695 The value of a function with a domain of two elements. (Moved to fvpr1 5621 in main set.mm and may be deleted by mathbox owner, JM. --NM 3-Sep-2011.) (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  A )  =  C )
 
Theoremprfv2OLD 25696 The value of a function with a domain of two elements. (Moved to fvpr2 5622 in main set.mm and may be deleted by mathbox owner, JM. --NM 3-Sep-2011.) (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  B )  =  D )
 
TheoremprfOLD 25697 A function with a domain of two elements. (Moved to fpr 5603 in main set.mm and may be deleted by mathbox owner, JM. --NM 3-Sep-2011.) (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  { <. A ,  C >. ,  <. B ,  D >. } : { A ,  B } --> { C ,  D }
 )
 
Theorembrabg2 25698* Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   &    |-  ( ch  ->  A  e.  C )   =>    |-  ( B  e.  D  ->  ( A R B  <->  ch ) )
 
TheoreminpreimaOLD 25699 Preimage of an intersection. (Moved to inpreima 5551 in main set.mm and may be deleted by mathbox owner, JM. --NM 28-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Fun  F  ->  ( `' F " ( A  i^i  B ) )  =  ( ( `' F " A )  i^i  ( `' F " B ) ) )
 
TheoremunpreimaOLD 25700 Preimage of a union. (Moved to unpreima 5550 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Fun  F  ->  ( `' F " ( A  u.  B ) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
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