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Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfnerel 26301 Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  Rel  Fne
 
Theoremisfne 26302* The predicate " B is finer than  A." This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x  C_ 
 U. ( B  i^i  ~P x ) ) ) )
 
Theoremisfne4 26303 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 26304 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 26305* 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 26306* 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 26307 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 26308 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 26309* 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 26310* 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 26311* 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 26312 Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  Rel  Ref
 
Theoremisref 26313* 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 26302. 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 26314 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 26315* 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 26316 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 26317 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 26318 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
 |-  ( A  e.  V  ->  A Fne A )
 
Theoremrefref 26319 Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  ( A  e.  V  ->  A Ref A )
 
Theoremfnetr 26320 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 26321 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 26322 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 26323 Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  (
 ( A Ref B  /\  B Ref C ) 
 ->  A Ref C )
 
Theoremtopfne 26324 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 26325 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 26326 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 26327* 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 26328* 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 26329* 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 26330* 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 26331 A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  ( A  e.  Fin  ->  A  e.  PtFin )
 
Theoremptfinfin 26332* 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 26333 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 26334 A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  ( A  e.  ( LocFin `  J )  ->  J  e.  Top )
 
Theoremlocfinbas 26335 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 26336* 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 26337 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 26338 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 26339 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 26340 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 26341* 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 ) ) ) )
 
19.13.5  Neighborhood bases determine topologies
 
Theoremneibastop1 26342* 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 26343* Lemma for neibastop2 26344. (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 26344* 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 26345* 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 )  =/=  (/) } )
 
19.13.6  Lattice structure of topologies
 
Theoremtopmtcl 26346 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 26347* 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 26348* 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 26349* 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 26350* 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 26351* 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 26352* 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 ) ) )
 
19.13.7  Filter bases
 
Theoremfgmin 26353 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 26354* The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 17864. (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 ) )
 
19.13.8  Directed sets, nets
 
Theoremtailfval 26355* 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 26356 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 26357 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 26358 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 26359 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 26360 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 26361* Lemma for filnet 26365. 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 26362* Lemma for filnet 26365. 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 26363* Lemma for filnet 26365. (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 26364* Lemma for filnet 26365. (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 26365* 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 )
 ) ) )
 
19.14  Mathbox for Jeff Madsen
 
19.14.1  Logic and set theory
 
Theoremanim12da 26366 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 )
 )
 
Theoremsyldanl 26367 A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   &    |-  ( ( (
 ph  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ph  /\  ps )  /\  th )  ->  ta )
 
Theoremunirep 26368* 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 )
 
Theoremcover2 26369* 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 26370* 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 ) ) )
 
Theorembrabg2 26371* 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 ) )
 
Theoremopelopab3 26372* Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  ( ch  ->  A  e.  C )   =>    |-  ( B  e.  D  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch ) )
 
Theoremcocanfo 26373 Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  (
 ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B )  /\  ( G  o.  F )  =  ( H  o.  F ) ) 
 ->  G  =  H )
 
Theorembrresi 26374 Restriction of a binary relation. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  B  e.  _V   =>    |-  ( A ( R  |`  C ) B  ->  A R B )
 
Theoremfnopabeqd 26375* Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) } )
 
Theoremfvopabf4g 26376* Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  C  e.  _V   &    |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  ( R 
 ^m  D )  |->  B )   =>    |-  ( ( D  e.  X  /\  R  e.  Y  /\  A : D --> R ) 
 ->  ( F `  A )  =  C )
 
Theoremeqfnun 26377 Two functions on  A  u.  B are equal if and only if they have equal restrictions to both  A and  B. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  (
 ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  ->  ( F  =  G  <->  ( ( F  |`  A )  =  ( G  |`  A ) 
 /\  ( F  |`  B )  =  ( G  |`  B ) ) ) )
 
Theoremfnopabco 26378* Composition of a function with a function abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  ( x  e.  A  ->  B  e.  C )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }   &    |-  G  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( H `  B ) ) }   =>    |-  ( H  Fn  C  ->  G  =  ( H  o.  F ) )
 
Theoremopropabco 26379* Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
 |-  ( x  e.  A  ->  B  e.  R )   &    |-  ( x  e.  A  ->  C  e.  S )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  = 
 <. B ,  C >. ) }   &    |-  G  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( B M C ) ) }   =>    |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
 
Theoremf1opr 26380* Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( F : ( A  X.  B ) -1-1-> C  <->  ( F :
 ( A  X.  B )
 --> C  /\  A. r  e.  A  A. s  e.  B  A. t  e.  A  A. u  e.  B  ( ( r F s )  =  ( t F u )  ->  ( r  =  t  /\  s  =  u ) ) ) )
 
Theoremcocnv 26381 Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( Fun  F  /\  Fun 
 G )  ->  (
 ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
 
Theoremf1ocan1fv 26382 Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  (
 ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  ( ( F  o.  G ) `
  ( `' G `  X ) )  =  ( F `  X ) )
 
Theoremf1ocan2fv 26383 Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  ( ( F  o.  `' G ) `  ( G `  X ) )  =  ( F `  X ) )
 
Theoreminixp 26384* Intersection of Cartesian products over the same base set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( X_ x  e.  A  B  i^i  X_ x  e.  A  C )  =  X_ x  e.  A  ( B  i^i  C )
 
Theoremupixp 26385* Universal property of the indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  X  =  X_ b  e.  A  ( C `  b )   &    |-  P  =  ( w  e.  A  |->  ( x  e.  X  |->  ( x `  w ) ) )   =>    |-  ( ( A  e.  R  /\  B  e.  S  /\  A. a  e.  A  ( F `  a ) : B --> ( C `
  a ) ) 
 ->  E! h ( h : B --> X  /\  A. a  e.  A  ( F `  a )  =  ( ( P `
  a )  o.  h ) ) )
 
Theoremabrexdom 26386* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 y  e.  A  ->  E* x ph )   =>    |-  ( A  e.  V  ->  { x  |  E. y  e.  A  ph
 }  ~<_  A )
 
Theoremabrexdom2 26387* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  V  ->  { x  |  E. y  e.  A  x  =  B } 
 ~<_  A )
 
Theoremac6gf 26388* Axiom of Choice. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  F/ y ps   &    |-  ( y  =  ( f `  x )  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  C  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremindexa 26389* If for every element of an indexing set  A there exists a corresponding element of another set  B, then there exists a subset of  B consisting only of those elements which are indexed by  A. Used to avoid the Axiom of Choice in situations where only the range of the choice function is needed. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( B  e.  M  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. c
 ( c  C_  B  /\  A. x  e.  A  E. y  e.  c  ph 
 /\  A. y  e.  c  E. x  e.  A  ph ) )
 
Theoremindexdom 26390* If for every element of an indexing set  A there exists a corresponding element of another set  B, then there exists a subset of  B consisting only of those elements which are indexed by  A, and which is dominated by the set  A. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( A  e.  M  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. c
 ( ( c  ~<_  A 
 /\  c  C_  B )  /\  ( A. x  e.  A  E. y  e.  c  ph  /\  A. y  e.  c  E. x  e.  A  ph ) ) )
 
Theoremfrinfm 26391* A subset of a well-founded set has an infimum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( R  Fr  A  /\  ( B  e.  C  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  A  ( A. y  e.  B  -.  x `' R y 
 /\  A. y  e.  A  ( y `' R x  ->  E. z  e.  B  y `' R z ) ) )
 
Theoremwelb 26392* A non-empty subset of a well-ordered set has a lower bound. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( R  We  A  /\  ( B  e.  C  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  ( `' R  Or  B  /\  E. x  e.  B  ( A. y  e.  B  -.  x `' R y 
 /\  A. y  e.  B  ( y `' R x  ->  E. z  e.  B  y `' R z ) ) ) )
 
Theoremsupex2g 26393 Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  C  ->  sup ( B ,  A ,  R )  e.  _V )
 
Theoremsupclt 26394* Closure of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( R  Or  A  /\  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )  ->  sup ( B ,  A ,  R )  e.  A )
 
Theoremsupubt 26395* Upper bound property of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( R  Or  A  /\  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )  ->  ( C  e.  B  ->  -.  sup ( B ,  A ,  R ) R C ) )
 
19.14.2  Real and complex numbers; integers
 
Theoremfilbcmb 26396* Combine a finite set of lower bounds. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( A  e.  Fin  /\  A  =/=  (/)  /\  B  C_ 
 RR )  ->  ( A. x  e.  A  E. y  e.  B  A. z  e.  B  ( y  <_  z  ->  ph )  ->  E. y  e.  B  A. z  e.  B  ( y  <_  z  ->  A. x  e.  A  ph ) ) )
 
Theoremrdr 26397 Two ways of expressing the remainder when  A is divided by 
B. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  -  ( A  mod  B ) )  /  B )  =  ( |_ `  ( A  /  B ) ) )
 
Theoremfzmul 26398 Membership of a product in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  NN )  ->  ( J  e.  ( M ... N )  ->  ( K  x.  J )  e.  ( ( K  x.  M ) ... ( K  x.  N ) ) ) )
 
Theoremfzadd2 26399 Membership of a sum in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  (
 ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( O  e.  ZZ  /\  P  e.  ZZ ) )  ->  ( ( J  e.  ( M
 ... N )  /\  K  e.  ( O ... P ) )  ->  ( J  +  K )  e.  ( ( M  +  O ) ... ( N  +  P ) ) ) )
 
19.14.3  Sequences and sums
 
Theoremsdclem2 26400* Lemma for sdc 26402. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  (
 g  =  ( f  |`  ( M ... n ) )  ->  ( ps  <->  ch ) )   &    |-  ( n  =  M  ->  ( ps  <->  ta ) )   &    |-  ( n  =  k  ->  ( ps  <->  th ) )   &    |-  ( ( g  =  h  /\  n  =  ( k  +  1 ) )  ->  ( ps 
 <-> 
 si ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. g ( g : { M } --> A  /\  ta ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( ( g : ( M ... k
 ) --> A  /\  th )  ->  E. h ( h : ( M ... ( k  +  1
 ) ) --> A  /\  g  =  ( h  |`  ( M ... k
 ) )  /\  si ) ) )   &    |-  J  =  { g  |  E. n  e.  Z  (
 g : ( M
 ... n ) --> A  /\  ps ) }   &    |-  F  =  ( w  e.  Z ,  x  e.  J  |->  { h  |  E. k  e.  Z  ( h : ( M
 ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k
 ) )  /\  si ) } )   &    |-  F/ k ph   &    |-  ( ph  ->  G : Z --> J )   &    |-  ( ph  ->  ( G `  M ) : ( M ... M ) --> A )   &    |-  (
 ( ph  /\  w  e.  Z )  ->  ( G `  ( w  +  1 ) )  e.  ( w F ( G `  w ) ) )   =>    |-  ( ph  ->  E. f
 ( f : Z --> A  /\  A. n  e.  Z  ch ) )
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