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Theorem List for Metamath Proof Explorer - 17601-17700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremufilb 17601 The complement is in an ultrafilter iff the set is not. (Contributed by Mario Carneiro, 11-Dec-2013.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( -.  S  e.  F 
 <->  ( X  \  S )  e.  F )
 )
 
Theoremufilmax 17602 Any filter finer than an ultrafilter is actually equal to it. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  ->  F  =  G )
 
Theoremisufil2 17603* The maximal property of an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  <->  ( F  e.  ( Fil `  X )  /\  A. f  e.  ( Fil `  X ) ( F  C_  f  ->  F  =  f ) ) )
 
Theoremufprim 17604 An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  A  C_  X  /\  B  C_  X )  ->  ( ( A  e.  F  \/  B  e.  F ) 
 <->  ( A  u.  B )  e.  F )
 )
 
Theoremtrufil 17605 Conditions for the trace of an ultrafilter  L to be an ultrafilter. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ( L  e.  ( UFil `  Y )  /\  A  C_  Y )  ->  ( ( Lt  A )  e.  ( UFil `  A ) 
 <->  A  e.  L ) )
 
Theoremfilssufilg 17606* A filter is contained in some ultrafilter. This version of filssufil 17607 contains the choice as a hypothesis (in the assumption that  ~P ~P X is well-orderable). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  ~P ~P X  e.  dom  card )  ->  E. f  e.  ( UFil `  X ) F  C_  f )
 
Theoremfilssufil 17607* A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 8097.) (Contributed by Jeff Hankins, 2-Dec-2009.) (Revised by Stefan O'Rear, 29-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  E. f  e.  ( UFil `  X ) F 
 C_  f )
 
Theoremisufl 17608* Define the (strong) ultrafilter lemma, parameterized over base sets. A set  X satisfies the ultrafilter lemma if every filter on  X is a subset of some ultrafilter. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( X  e.  V  ->  ( X  e. UFL  <->  A. f  e.  ( Fil `  X ) E. g  e.  ( UFil `  X ) f  C_  g ) )
 
Theoremufli 17609* Property of a set that satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( X  e. UFL  /\  F  e.  ( Fil `  X ) )  ->  E. f  e.  ( UFil `  X ) F 
 C_  f )
 
Theoremnumufl 17610 Consequence of filssufilg 17606: a set whose double powerset is well-orderable satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ~P ~P X  e.  dom  card  ->  X  e. UFL )
 
Theoremfiufl 17611 A finite set satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( X  e.  Fin  ->  X  e. UFL )
 
Theoremacufl 17612 The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  (CHOICE 
 -> UFL  =  _V )
 
Theoremssufl 17613 If  Y is a subset of  X and filters extend to ultrafilters in  X, then they still do in  Y. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( X  e. UFL  /\  Y  C_  X )  ->  Y  e. UFL )
 
Theoremufileu 17614* If the ultrafilter containing a given filter is unique, the filter is an ultrafilter. (Contributed by Jeff Hankins, 3-Dec-2009.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( F  e.  ( UFil `  X )  <->  E! f  e.  ( UFil `  X ) F 
 C_  f ) )
 
Theoremfilufint 17615* A filter is equal to the intersection of the ultrafilters containing it. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  |^| { f  e.  ( UFil `  X )  |  F  C_  f }  =  F )
 
Theoremuffix 17616* Lemma for fixufil 17617 and uffixfr 17618. (Contributed by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( X  e.  V  /\  A  e.  X )  ->  ( { { A } }  e.  ( fBas `  X )  /\  { x  e.  ~P X  |  A  e.  x }  =  ( X filGen { { A } }
 ) ) )
 
Theoremfixufil 17617* The condition describing a fixed ultrafilter always produces an ultrafilter. (Contributed by Jeff Hankins, 9-Dec-2009.) (Revised by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 29-Jul-2015.)
 |-  ( ( X  e.  V  /\  A  e.  X )  ->  { x  e. 
 ~P X  |  A  e.  x }  e.  ( UFil `  X ) )
 
Theoremuffixfr 17618* An ultrafilter is either fixed or free. A fixed ultrafilter is called principal (generated by a single element  A), and a free ultrafilter is called nonprincipal (having empty intersection). Note that examples of free ultrafilters cannot be defined in ZFC without some form of global choice. (Contributed by Jeff Hankins, 4-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  ( A  e.  |^| F  <->  F  =  { x  e. 
 ~P X  |  A  e.  x } ) )
 
Theoremuffix2 17619* A classification of fixed ultrafilters. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  ( |^| F  =/=  (/)  <->  E. x  e.  X  F  =  { y  e.  ~P X  |  x  e.  y } ) )
 
Theoremuffixsn 17620 The singleton of the generator of a fixed ultrafilter is in the filter. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  A  e.  |^| F )  ->  { A }  e.  F )
 
Theoremufildom1 17621 An ultrafilter is generated by at most one element (because free ultrafilters have no generators and fixed ultrafilters have exactly one). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  |^| F  ~<_  1o )
 
Theoremuffinfix 17622* An ultrafilter containing a finite element is fixed. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  S  e.  F  /\  S  e.  Fin )  ->  E. x  e.  X  F  =  { y  e.  ~P X  |  x  e.  y } )
 
Theoremcfinufil 17623* An ultrafilter is free iff it contains the Fréchet filter cfinfil 17588 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  ( |^| F  =  (/)  <->  { x  e.  ~P X  |  ( X  \  x )  e.  Fin }  C_  F ) )
 
Theoremufinffr 17624* An infinite subset is contained in a free ultrafilter. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Mario Carneiro, 4-Dec-2013.)
 |-  ( ( X  e.  B  /\  A  C_  X  /\  om  ~<_  A )  ->  E. f  e.  ( UFil `  X ) ( A  e.  f  /\  |^| f  =  (/) ) )
 
Theoremufilen 17625* Any infinite set has an ultrafilter on it whose elements are of the same cardinality as the set. Any such ultrafilter is necessarily free. (Contributed by Jeff Hankins, 7-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)
 |-  ( om  ~<_  X  ->  E. f  e.  ( UFil `  X ) A. x  e.  f  x  ~~  X )
 
Theoremufildr 17626 An ultrafilter gives rise to a connected door topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)
 |-  J  =  ( F  u.  { (/) } )   =>    |-  ( F  e.  ( UFil `  X )  ->  ( J  u.  ( Clsd `  J ) )  =  ~P X )
 
Theoremfin1aufil 17627 There are no definable free ultrafilters in ZFC. However, there are free ultrafilters in some choice-denying constructions. Here we show that given an amorphous set (a.k.a. a Ia-finite I-infinite set)  X, the set of infinite subsets of 
X is a free ultrafilter on  X. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  F  =  ( ~P X  \  Fin )   =>    |-  ( X  e.  (FinIa  \  Fin )  ->  ( F  e.  ( UFil `  X )  /\  |^| F  =  (/) ) )
 
11.2.4  Filter limits
 
Syntaxcfm 17628 Extend class definition to include the neighborhood filter mapping function.
 class  FilMap
 
Syntaxcflim 17629 Extend class notation with a function returning the limit of a filter.
 class  fLim
 
Syntaxcflf 17630 Extend class definition to include the function for filter-based function limits.
 class  fLimf
 
Syntaxcfcls 17631 Extend class definition to include the cluster point function on filters.
 class  fClus
 
Syntaxcfcf 17632 Extend class definition to include the function for cluster points of a function.
 class  fClusf
 
Definitiondf-fm 17633* Define a function that takes a filter to a neighborhood filter of the range. (Since we now allow filter bases to have support smaller than the base set, the function has to come first to ensure that curryings are sets.) (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 20-Jul-2015.)
 |-  FilMap  =  ( x  e. 
 _V ,  f  e. 
 _V  |->  ( y  e.  ( fBas `  dom  f ) 
 |->  ( x filGen ran  (
 t  e.  y  |->  ( f " t ) ) ) ) )
 
Definitiondf-flim 17634* Define a function (indexed by a topology  j) whose value is the limits of a filter  f. (Contributed by Jeff Hankins, 4-Sep-2009.)
 |- 
 fLim  =  ( j  e.  Top ,  f  e. 
 U. ran  Fil  |->  { x  e.  U. j  |  ( ( ( nei `  j
 ) `  { x } )  C_  f  /\  f  C_  ~P U. j
 ) } )
 
Definitiondf-flf 17635* Define a function that gives the limits of a function  f in the filter sense. (Contributed by Jeff Hankins, 14-Oct-2009.)
 |- 
 fLimf  =  ( x  e.  Top ,  y  e. 
 U. ran  Fil  |->  ( f  e.  ( U. x  ^m  U. y )  |->  ( x  fLim  ( ( U. x  FilMap  f ) `
  y ) ) ) )
 
Definitiondf-fcls 17636* Define a function that takes a filter in a topology to its set of cluster points. (Contributed by Jeff Hankins, 10-Nov-2009.)
 |- 
 fClus  =  ( j  e.  Top ,  f  e. 
 U. ran  Fil  |->  if ( U. j  =  U. f ,  |^|_ x  e.  f  ( ( cls `  j ) `  x ) ,  (/) ) )
 
Definitiondf-fcf 17637* Define a function that gives the cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.)
 |-  fClusf  =  ( j  e. 
 Top ,  f  e.  U.
 ran  Fil  |->  ( g  e.  ( U. j  ^m  U. f )  |->  ( j 
 fClus  ( ( U. j  FilMap  g ) `  f
 ) ) ) )
 
Theoremfmval 17638* Introduce a function that takes a function from a filtered domain to a set and produces a filter which consists of supersets of images of filter elements. The functions which are dealt with by this function are similar to nets in topology. For example, suppose we have a sequence filtered by the filter generated by its tails under the usual natural number ordering. Then the elements of this filter are precisely the supersets of tails of this sequence. Under this definition, it is not too difficult to see that the limit of a function in the filter sense captures the notion of convergence of a sequence. As a result, the notion of a filter generalizes many ideas associated with sequences, and this function is one way to make that relationship precise in Metamath. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X ) 
 ->  ( ( X  FilMap  F ) `  B )  =  ( X filGen ran  ( y  e.  B  |->  ( F " y ) ) ) )
 
Theoremfmfil 17639 A mapping filter is a filter. (Contributed by Jeff Hankins, 18-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X ) 
 ->  ( ( X  FilMap  F ) `  B )  e.  ( Fil `  X ) )
 
Theoremfmf 17640 Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( X  e.  A  /\  Y  e.  B  /\  F : Y --> X ) 
 ->  ( X  FilMap  F ) : ( fBas `  Y )
 --> ( Fil `  X ) )
 
Theoremfmss 17641 A finer filter produces a finer image filter. (Contributed by Jeff Hankins, 16-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  C  e.  ( fBas `  Y ) )  /\  ( F : Y --> X  /\  B  C_  C ) ) 
 ->  ( ( X  FilMap  F ) `  B ) 
 C_  ( ( X 
 FilMap  F ) `  C ) )
 
Theoremelfm 17642* An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X ) 
 ->  ( A  e.  (
 ( X  FilMap  F ) `
  B )  <->  ( A  C_  X  /\  E. x  e.  B  ( F " x )  C_  A ) ) )
 
Theoremelfm2 17643* An element of a mapping filter. (Contributed by Jeff Hankins, 26-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  L  =  ( Y
 filGen B )   =>    |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X ) 
 ->  ( A  e.  (
 ( X  FilMap  F ) `
  B )  <->  ( A  C_  X  /\  E. x  e.  L  ( F " x )  C_  A ) ) )
 
Theoremfmfg 17644 The image filter of a filter base is the same as the image filter of its generated filter. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  L  =  ( Y
 filGen B )   =>    |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X ) 
 ->  ( ( X  FilMap  F ) `  B )  =  ( ( X 
 FilMap  F ) `  L ) )
 
Theoremelfm3 17645* An alternate formulation of elementhood in a mapping filter that requires  F to be onto. (Contributed by Jeff Hankins, 1-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  L  =  ( Y
 filGen B )   =>    |-  ( ( B  e.  ( fBas `  Y )  /\  F : Y -onto-> X )  ->  ( A  e.  ( ( X  FilMap  F ) `  B )  <->  E. x  e.  L  A  =  ( F " x ) ) )
 
Theoremimaelfm 17646 An image of a filter element is in the image filter. (Contributed by Jeff Hankins, 5-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  L  =  ( Y
 filGen B )   =>    |-  ( ( ( X  e.  A  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X ) 
 /\  S  e.  L )  ->  ( F " S )  e.  (
 ( X  FilMap  F ) `
  B ) )
 
Theoremrnelfmlem 17647* Lemma for rnelfm 17648. (Contributed by Jeff Hankins, 14-Nov-2009.)
 |-  ( ( ( Y  e.  A  /\  L  e.  ( Fil `  X )  /\  F : Y --> X )  /\  ran  F  e.  L )  ->  ran  ( x  e.  L  |->  ( `' F " x ) )  e.  ( fBas `  Y ) )
 
Theoremrnelfm 17648 A condition for a filter to be an image filter for a given function. (Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( Y  e.  A  /\  L  e.  ( Fil `  X )  /\  F : Y --> X ) 
 ->  ( L  e.  ran  ( X  FilMap  F )  <->  ran  F  e.  L ) )
 
Theoremfmfnfmlem1 17649* Lemma for fmfnfm 17653. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ph  ->  B  e.  ( fBas `  Y )
 )   &    |-  ( ph  ->  L  e.  ( Fil `  X ) )   &    |-  ( ph  ->  F : Y --> X )   &    |-  ( ph  ->  ( ( X  FilMap  F ) `  B )  C_  L )   =>    |-  ( ph  ->  ( s  e.  ( fi `  B )  ->  ( ( F
 " s )  C_  t  ->  ( t  C_  X  ->  t  e.  L ) ) ) )
 
Theoremfmfnfmlem2 17650* Lemma for fmfnfm 17653. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ph  ->  B  e.  ( fBas `  Y )
 )   &    |-  ( ph  ->  L  e.  ( Fil `  X ) )   &    |-  ( ph  ->  F : Y --> X )   &    |-  ( ph  ->  ( ( X  FilMap  F ) `  B )  C_  L )   =>    |-  ( ph  ->  ( E. x  e.  L  s  =  ( `' F " x )  ->  ( ( F " s ) 
 C_  t  ->  (
 t  C_  X  ->  t  e.  L ) ) ) )
 
Theoremfmfnfmlem3 17651* Lemma for fmfnfm 17653. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ph  ->  B  e.  ( fBas `  Y )
 )   &    |-  ( ph  ->  L  e.  ( Fil `  X ) )   &    |-  ( ph  ->  F : Y --> X )   &    |-  ( ph  ->  ( ( X  FilMap  F ) `  B )  C_  L )   =>    |-  ( ph  ->  ( fi ` 
 ran  ( x  e.  L  |->  ( `' F " x ) ) )  =  ran  ( x  e.  L  |->  ( `' F " x ) ) )
 
Theoremfmfnfmlem4 17652* Lemma for fmfnfm 17653. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ph  ->  B  e.  ( fBas `  Y )
 )   &    |-  ( ph  ->  L  e.  ( Fil `  X ) )   &    |-  ( ph  ->  F : Y --> X )   &    |-  ( ph  ->  ( ( X  FilMap  F ) `  B )  C_  L )   =>    |-  ( ph  ->  ( t  e.  L  <->  ( t  C_  X  /\  E. s  e.  ( fi `  ( B  u.  ran  ( x  e.  L  |->  ( `' F " x ) ) ) ) ( F "
 s )  C_  t
 ) ) )
 
Theoremfmfnfm 17653* A filter finer than an image filter is an image filter of the same function. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ph  ->  B  e.  ( fBas `  Y )
 )   &    |-  ( ph  ->  L  e.  ( Fil `  X ) )   &    |-  ( ph  ->  F : Y --> X )   &    |-  ( ph  ->  ( ( X  FilMap  F ) `  B )  C_  L )   =>    |-  ( ph  ->  E. f  e.  ( Fil `  Y ) ( B  C_  f  /\  L  =  ( ( X  FilMap  F ) `
  f ) ) )
 
Theoremfmufil 17654 An image filter of an ultrafilter is an ultrafilter. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( X  e.  A  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X ) 
 ->  ( ( X  FilMap  F ) `  L )  e.  ( UFil `  X ) )
 
Theoremfmid 17655 The filter map applied to the identity. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Mario Carneiro, 27-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( ( X  FilMap  (  _I  |`  X )
 ) `  F )  =  F )
 
Theoremfmco 17656 Composition of image filters. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ( ( X  e.  V  /\  Y  e.  W  /\  B  e.  ( fBas `  Z )
 )  /\  ( F : Y --> X  /\  G : Z --> Y ) ) 
 ->  ( ( X  FilMap  ( F  o.  G ) ) `  B )  =  ( ( X 
 FilMap  F ) `  (
 ( Y  FilMap  G ) `
  B ) ) )
 
Theoremufldom 17657 The ultrafilter lemma property is a cardinal invariant, so since it transfers to subsets it also transfers over set dominance. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( X  e. UFL  /\  Y  ~<_  X )  ->  Y  e. UFL )
 
Theoremflimval 17658* The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  ( J  fLim  F )  =  { x  e.  X  |  ( ( ( nei `  J ) `  { x } )  C_  F  /\  F  C_  ~P X ) } )
 
Theoremelflim2 17659 The predicate "is a limit point of a filter." (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( A  e.  ( J  fLim  F )  <->  ( ( J  e.  Top  /\  F  e.  U.
 ran  Fil  /\  F  C_  ~P X )  /\  ( A  e.  X  /\  ( ( nei `  J ) `  { A } )  C_  F ) ) )
 
Theoremflimtop 17660 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
 
Theoremflimneiss 17661 A filter contains the neighborhood filter as a subfilter. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( A  e.  ( J  fLim  F )  ->  ( ( nei `  J ) `  { A }
 )  C_  F )
 
Theoremflimnei 17662 A filter contains all of the neighborhoods of its limit points. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.)
 |-  ( ( A  e.  ( J  fLim  F ) 
 /\  N  e.  (
 ( nei `  J ) `  { A } )
 )  ->  N  e.  F )
 
Theoremflimelbas 17663 A limit point of a filter belongs to its base set. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   =>    |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )
 
Theoremflimfil 17664 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  X ) )
 
Theoremflimtopon 17665 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X ) ) )
 
Theoremelflim 17666 The predicate "is a limit point of a filter." (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X ) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  ( ( nei `  J ) `  { A } )  C_  F ) ) )
 
Theoremflimss2 17667 A limit point of a filter is a limit point of a finer filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X )  /\  G  C_  F )  ->  ( J  fLim  G )  C_  ( J  fLim  F ) )
 
Theoremflimss1 17668 A limit point of a filter is a limit point in a coarser topology. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X )  /\  J  C_  K )  ->  ( K  fLim  F )  C_  ( J  fLim  F ) )
 
Theoremneiflim 17669 A point is a limit point of its neighborhood filter. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( J 
 fLim  ( ( nei `  J ) `  { A }
 ) ) )
 
Theoremflimopn 17670* The condition for being a limit point of a filter still holds if one only considers open neighborhoods. (Contributed by Jeff Hankins, 4-Sep-2009.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X ) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  A. x  e.  J  ( A  e.  x  ->  x  e.  F ) ) ) )
 
Theoremfbflim 17671* A condition for a filter to converge to a point involving one of its bases. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  F  =  ( X
 filGen B )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  A. x  e.  J  ( A  e.  x  ->  E. y  e.  B  y  C_  x ) ) ) )
 
Theoremfbflim2 17672* A condition for a filter base  B to converge to a point 
A. Use neighborhoods instead of open neighborhoods. Compare fbflim 17671. (Contributed by FL, 4-Jul-2011.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  F  =  ( X
 filGen B )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J ) `  { A }
 ) E. x  e.  B  x  C_  n ) ) )
 
Theoremflimclsi 17673 The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( S  e.  F  ->  ( J  fLim  F ) 
 C_  ( ( cls `  J ) `  S ) )
 
Theoremhausflimlem 17674 If  A and  B are both limits of the same filter, then all neighborhoods of  A and  B intersect. (Contributed by Mario Carneiro, 21-Sep-2015.)
 |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
 )  ->  ( U  i^i  V )  =/=  (/) )
 
Theoremhausflimi 17675* One direction of hausflim 17676. A filter in a Hausdorf space has at most one limit. (Contributed by FL, 14-Nov-2010.) (Revised by Mario Carneiro, 21-Sep-2015.)
 |-  ( J  e.  Haus  ->  E* x  x  e.  ( J  fLim  F ) )
 
Theoremhausflim 17676* A condition for a topology to be Hausdorff in terms of filters. A topology is Hausdorff iff every filter has at most one limit point. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Haus  <->  ( J  e.  Top  /\  A. f  e.  ( Fil `  X ) E* x  x  e.  ( J  fLim  f ) ) )
 
Theoremflimcf 17677* Fineness is properly characterized by the property that every limit point of a filter in the finer topology is a limit point in the coarser topology. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  ->  ( J 
 C_  K  <->  A. f  e.  ( Fil `  X ) ( K  fLim  f )  C_  ( J  fLim  f
 ) ) )
 
Theoremflimrest 17678 The set of limit points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X )  /\  Y  e.  F )  ->  (
 ( Jt  Y )  fLim  ( Ft  Y ) )  =  ( ( J  fLim  F )  i^i  Y ) )
 
Theoremflimclslem 17679 Lemma for flimcls 17680. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  F  =  ( X
 filGen ( fi `  (
 ( ( nei `  J ) `  { A }
 )  u.  { S } ) ) )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  A  e.  ( ( cls `  J ) `  S ) ) 
 ->  ( F  e.  ( Fil `  X )  /\  S  e.  F  /\  A  e.  ( J  fLim  F ) ) )
 
Theoremflimcls 17680* Closure in terms of filter convergence. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  ( A  e.  (
 ( cls `  J ) `  S )  <->  E. f  e.  ( Fil `  X ) ( S  e.  f  /\  A  e.  ( J  fLim  f ) ) ) )
 
Theoremflimsncls 17681 If  A is a limit point of the filter  F, then all the points which specialize  A (in the specialization preorder) are also limit points. Thus the set of limit points is a union of closed sets (although this is only nontrivial for non-T1 spaces). (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  ( A  e.  ( J  fLim  F )  ->  ( ( cls `  J ) `  { A }
 )  C_  ( J  fLim  F ) )
 
Theoremhauspwpwf1 17682* Lemma for hauspwpwdom 17683. Points in the closure of a set in a Hausdorff space are characterized by the open neighborhoods they extend into the generating set. (Contributed by Mario Carneiro, 28-Jul-2015.)
 |-  X  =  U. J   &    |-  F  =  ( x  e.  (
 ( cls `  J ) `  A )  |->  { a  |  E. j  e.  J  ( x  e.  j  /\  a  =  (
 j  i^i  A )
 ) } )   =>    |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  F :
 ( ( cls `  J ) `  A ) -1-1-> ~P ~P A )
 
Theoremhauspwpwdom 17683 If  X is a Hausdorff space, then the cardinality of the closure of a set  A is bounded by the double powerset of  A. In particular, a Hausdorff space with a dense subset  A has cardinality at most  ~P ~P A, and a separable Hausdorff space has cardinality at most  ~P ~P NN. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Haus  /\  A  C_  X )  ->  ( ( cls `  J ) `  A )  ~<_  ~P
 ~P A )
 
Theoremflffval 17684* Given a topology and a filtered set, return the convergence function on the functions from the filtered set to the base set of the topological space. (Contributed by Jeff Hankins, 14-Oct-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y ) )  ->  ( J  fLimf  L )  =  ( f  e.  ( X  ^m  Y )  |->  ( J  fLim  ( ( X  FilMap  f ) `
  L ) ) ) )
 
Theoremflfval 17685 Given a function from a filtered set to a topological space, define the set of limit points of the function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( ( J  fLimf  L ) `  F )  =  ( J  fLim  ( ( X  FilMap  F ) `
  L ) ) )
 
Theoremflfnei 17686* The property of being a limit point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J ) `  { A }
 ) E. s  e.  L  ( F "
 s )  C_  n ) ) )
 
Theoremflfneii 17687* A neighborhood of a limit point of a function contains the image of a filter element. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  A  e.  ( ( J  fLimf  L ) `  F )  /\  N  e.  ( ( nei `  J ) `  { A }
 ) )  ->  E. s  e.  L  ( F "
 s )  C_  N )
 
Theoremisflf 17688* The property of being a limit point of a function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  E. s  e.  L  ( F " s ) 
 C_  o ) ) ) )
 
Theoremflfelbas 17689 A limit point of a function is in the topological space. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  A  e.  (
 ( J  fLimf  L ) `
  F ) ) 
 ->  A  e.  X )
 
Theoremflffbas 17690* Limit points of a function can be defined using filter bases. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  L  =  ( Y
 filGen B )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  E. s  e.  B  ( F " s ) 
 C_  o ) ) ) )
 
Theoremflftg 17691* Limit points of a function can be defined using topological bases. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( topGen `  B )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  <->  ( A  e.  X  /\  A. o  e.  B  ( A  e.  o  ->  E. s  e.  L  ( F " s ) 
 C_  o ) ) ) )
 
Theoremhausflf 17692* If a function has its values in a Hausdorff space then it has at most one limit value. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  E* x  x  e.  ( ( J  fLimf  L ) `  F ) )
 
Theoremhausflf2 17693 If a convergent function has its values in a Hausdorff space then it has a unique limit. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( ( J  fLimf  L ) `  F )  =/=  (/) )  ->  (
 ( J  fLimf  L ) `
  F )  ~~  1o )
 
Theoremcnpflfi 17694 Forward direction of cnpflf 17696. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( A  e.  ( J  fLim  L ) 
 /\  F  e.  (
 ( J  CnP  K ) `  A ) ) 
 ->  ( F `  A )  e.  ( ( K  fLimf  L ) `  F ) )
 
Theoremcnpflf2 17695  F is continous at point  A iff a limit of  F when  x tends to  A is  ( F `  A ). Proposition 9 of [BourbakiTop1] p. TG I.50. (Contributed by FL, 29-May-2011.) (Revised by Mario Carneiro, 9-Apr-2015.)
 |-  L  =  ( ( nei `  J ) `  { A } )   =>    |-  (
 ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X ) 
 ->  ( F  e.  (
 ( J  CnP  K ) `  A )  <->  ( F : X
 --> Y  /\  ( F `
  A )  e.  ( ( K  fLimf  L ) `  F ) ) ) )
 
Theoremcnpflf 17696* Continuity of a function at a point in terms of filter limits. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
 ( F : X --> Y  /\  A. f  e.  ( Fil `  X ) ( A  e.  ( J  fLim  f ) 
 ->  ( F `  A )  e.  ( ( K  fLimf  f ) `  F ) ) ) ) )
 
Theoremcnflf 17697* A function is continuous iff it respects filter limits. (Contributed by Jeff Hankins, 6-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. f  e.  ( Fil `  X ) A. x  e.  ( J  fLim  f ) ( F `  x )  e.  ( ( K 
 fLimf  f ) `  F ) ) ) )
 
Theoremcnflf2 17698* A function is continuous iff it respects filter limits. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. f  e.  ( Fil `  X ) ( F "
 ( J  fLim  f
 ) )  C_  (
 ( K  fLimf  f ) `
  F ) ) ) )
 
Theoremflfcnp 17699 A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  ( A  e.  ( ( J  fLimf  L ) `  F ) 
 /\  G  e.  (
 ( J  CnP  K ) `  A ) ) )  ->  ( G `  A )  e.  (
 ( K  fLimf  L ) `
  ( G  o.  F ) ) )
 
Theoremlmflf 17700 The topological limit relation on functions can be written in terms of the filter limit along the filter generated by the upper integer sets. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  L  =  ( Z filGen ( ZZ>= " Z ) )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z --> X )  ->  ( F ( ~~> t `  J ) P  <->  P  e.  (
 ( J  fLimf  L ) `
  F ) ) )
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