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Theorem List for Metamath Proof Explorer - 17601-17700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfgtr 17601 If  A is a member of the filter, then truncating  F to  A and regenerating the behavior outside  A using 
filGen recovers the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F ) 
 ->  ( X filGen ( Ft  A ) )  =  F )
 
Theoremtrfg 17602 The trace operation and the 
filGen operation are inverses to one another in some sense, with  filGen growing the base set and ↾t shrinking it. See fgtr 17601 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( F  e.  ( Fil `  A )  /\  A  C_  X  /\  X  e.  V )  ->  ( ( X filGen F )t  A )  =  F )
 
Theoremtrnei 17603 The trace, over a set  A, of the filter of the neighborhoods of a point  P is a filter iff  P belongs to the closure of  A. (This is trfil2 17598 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  ( P  e.  ( ( cls `  J ) `  A )  <->  ( ( ( nei `  J ) `  { P } )t  A )  e.  ( Fil `  A ) ) )
 
Theoremcfinfil 17604* Relative complements of the finite parts of an infinite set is a filter. When  A  =  NN the set of the relative complements is called Frechet's filter and is used to define the concept of limit of a sequence. (Contributed by FL, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( X  e.  V  /\  A  C_  X  /\  -.  A  e.  Fin )  ->  { x  e. 
 ~P X  |  ( A  \  x )  e.  Fin }  e.  ( Fil `  X )
 )
 
Theoremcsdfil 17605* The set of all elements whose complement is dominated by the base set is a filter. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( X  e.  dom  card  /\  om  ~<_  X ) 
 ->  { x  e.  ~P X  |  ( X  \  x )  ~<  X }  e.  ( Fil `  X ) )
 
Theoremsupfil 17606* The supersets of a nonempty set which are also subsets of a given base set form a filter. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  { x  e.  ~P A  |  B  C_  x }  e.  ( Fil `  A ) )
 
Theoremzfbas 17607 The set of upper integer sets is a filter base on  ZZ, which corresponds to convergence of sequences on  ZZ. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |- 
 ran  ZZ>=  e.  ( fBas `  ZZ )
 
Theoremuzrest 17608 The restriction of the set of upper integers to an upper integer set is the set of upper integers based at a point above the cutoff. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  ( ran  ZZ>=t  Z )  =  (
 ZZ>= " Z ) )
 
Theoremuzfbas 17609 The set of upper integer sets based at a point in a fixed upper integer set like  NN is a filter base on  NN, which corresponds to convergence of sequences on 
NN. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  ( ZZ>= " Z )  e.  ( fBas `  Z )
 )
 
11.2.3  Ultrafilters
 
Syntaxcufil 17610 Extend class notation with the ultrafilters-on-a-set function.
 class  UFil
 
Syntaxcufl 17611 Extend class notation with the ultrafilter lemma.
 class UFL
 
Definitiondf-ufil 17612* Define the set of ultrafilters on a set. An ultrafilter is a filter that gives a definite result for every subset. (Contributed by Jeff Hankins, 30-Nov-2009.)
 |- 
 UFil  =  ( g  e.  _V  |->  { f  e.  ( Fil `  g )  | 
 A. x  e.  ~P  g ( x  e.  f  \/  ( g 
 \  x )  e.  f ) } )
 
Definitiondf-ufl 17613* Define the class of base sets for which the ultrafilter lemma filssufil 17623 holds. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |- UFL 
 =  { x  |  A. f  e.  ( Fil `  x ) E. g  e.  ( UFil `  x ) f  C_  g }
 
Theoremisufil 17614* The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( F  e.  ( UFil `  X )  <->  ( F  e.  ( Fil `  X )  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X 
 \  x )  e.  F ) ) )
 
Theoremufilfil 17615 An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  F  e.  ( Fil `  X ) )
 
Theoremufilss 17616 For any subset of the base set of an ultrafilter, either the set is in the ultrafilter or the complement is. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( S  e.  F  \/  ( X  \  S )  e.  F )
 )
 
Theoremufilb 17617 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 17618 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 17619* 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 17620 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 17621 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 17622* A filter is contained in some ultrafilter. This version of filssufil 17623 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 17623* A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 8113.) (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 17624* 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 17625* 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 17626 Consequence of filssufilg 17622: 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 17627 A finite set satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( X  e.  Fin  ->  X  e. UFL )
 
Theoremacufl 17628 The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  (CHOICE 
 -> UFL  =  _V )
 
Theoremssufl 17629 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 17630* 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 17631* 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 17632* Lemma for fixufil 17633 and uffixfr 17634. (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 17633* 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 17634* 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 17635* 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 17636 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 17637 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 17638* 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 17639* An ultrafilter is free iff it contains the Fréchet filter cfinfil 17604 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 17640* 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 17641* 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 17642 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 17643 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 17644 Extend class definition to include the neighborhood filter mapping function.
 class  FilMap
 
Syntaxcflim 17645 Extend class notation with a function returning the limit of a filter.
 class  fLim
 
Syntaxcflf 17646 Extend class definition to include the function for filter-based function limits.
 class  fLimf
 
Syntaxcfcls 17647 Extend class definition to include the cluster point function on filters.
 class  fClus
 
Syntaxcfcf 17648 Extend class definition to include the function for cluster points of a function.
 class  fClusf
 
Definitiondf-fm 17649* 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 17650* 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 17651* 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 17652* 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 17653* 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 17654* 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 17655 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 17656 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 17657 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 17658* 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 17659* 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 17660 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 17661* 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 17662 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 17663* Lemma for rnelfm 17664. (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 17664 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 17665* Lemma for fmfnfm 17669. (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 17666* Lemma for fmfnfm 17669. (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 17667* Lemma for fmfnfm 17669. (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 17668* Lemma for fmfnfm 17669. (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 17669* 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 17670 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 17671 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 17672 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 17673 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 17674* 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 17675 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 17676 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 17677 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 17678 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 17679 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 17680 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 17681 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 17682 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 17683 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 17684 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 17685 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 17686* 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 17687* 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 17688* A condition for a filter base  B to converge to a point 
A. Use neighborhoods instead of open neighborhoods. Compare fbflim 17687. (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 17689 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 17690 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 17691* One direction of hausflim 17692. 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 17692* 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 17693* 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 17694 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 17695 Lemma for flimcls 17696. (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 17696* 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 17697 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 17698* Lemma for hauspwpwdom 17699. 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 17699 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 17700* 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 ) ) ) )
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