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Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfiufl 17901 A finite set satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremacufl 17902 The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
CHOICE UFL

Theoremssufl 17903 If is a subset of and filters extend to ultrafilters in , then they still do in . (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL UFL

Theoremufileu 17904* 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.)

Theoremfilufint 17905* 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.)

Theoremuffix 17906* Lemma for fixufil 17907 and uffixfr 17908. (Contributed by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfixufil 17907* 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.)

Theoremuffixfr 17908* An ultrafilter is either fixed or free. A fixed ultrafilter is called principal (generated by a single element ), 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.)

Theoremuffix2 17909* A classification of fixed ultrafilters. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremuffixsn 17910 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.)

Theoremufildom1 17911 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.)

Theoremuffinfix 17912* An ultrafilter containing a finite element is fixed. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremcfinufil 17913* An ultrafilter is free iff it contains the Fréchet filter cfinfil 17878 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremufinffr 17914* An infinite subset is contained in a free ultrafilter. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Mario Carneiro, 4-Dec-2013.)

Theoremufilen 17915* 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.)

Theoremufildr 17916 An ultrafilter gives rise to a connected door topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)

Theoremfin1aufil 17917 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) , the set of infinite subsets of is a free ultrafilter on . (Contributed by Mario Carneiro, 20-May-2015.)
FinIa

11.2.4  Filter limits

Syntaxcfm 17918 Extend class definition to include the neighborhood filter mapping function.

Syntaxcflim 17919 Extend class notation with a function returning the limit of a filter.

Syntaxcflf 17920 Extend class definition to include the function for filter-based function limits.

Syntaxcfcls 17921 Extend class definition to include the cluster point function on filters.

Syntaxcfcf 17922 Extend class definition to include the function for cluster points of a function.

Definitiondf-fm 17923* 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.)

Definitiondf-flim 17924* Define a function (indexed by a topology ) whose value is the limits of a filter . (Contributed by Jeff Hankins, 4-Sep-2009.)

Definitiondf-flf 17925* Define a function that gives the limits of a function in the filter sense. (Contributed by Jeff Hankins, 14-Oct-2009.)

Definitiondf-fcls 17926* Define a function that takes a filter in a topology to its set of cluster points. (Contributed by Jeff Hankins, 10-Nov-2009.)

Definitiondf-fcf 17927* Define a function that gives the cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.)

Theoremfmval 17928* 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.)

Theoremfmfil 17929 A mapping filter is a filter. (Contributed by Jeff Hankins, 18-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremfmf 17930 Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.)

Theoremfmss 17931 A finer filter produces a finer image filter. (Contributed by Jeff Hankins, 16-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremelfm 17932* An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremelfm2 17933* An element of a mapping filter. (Contributed by Jeff Hankins, 26-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremfmfg 17934 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.)

Theoremelfm3 17935* An alternate formulation of elementhood in a mapping filter that requires to be onto. (Contributed by Jeff Hankins, 1-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremimaelfm 17936 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.)

Theoremrnelfmlem 17937* Lemma for rnelfm 17938. (Contributed by Jeff Hankins, 14-Nov-2009.)

Theoremrnelfm 17938 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.)

Theoremfmfnfmlem1 17939* Lemma for fmfnfm 17943. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfmlem2 17940* Lemma for fmfnfm 17943. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfmlem3 17941* Lemma for fmfnfm 17943. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfmlem4 17942* Lemma for fmfnfm 17943. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfm 17943* 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.)

Theoremfmufil 17944 An image filter of an ultrafilter is an ultrafilter. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmid 17945 The filter map applied to the identity. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Mario Carneiro, 27-Aug-2015.)

Theoremfmco 17946 Composition of image filters. (Contributed by Mario Carneiro, 27-Aug-2015.)

Theoremufldom 17947 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.)
UFL UFL

Theoremflimval 17948* The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremelflim2 17949 The predicate "is a limit point of a filter." (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremflimtop 17950 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Theoremflimneiss 17951 A filter contains the neighborhood filter as a subfilter. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Theoremflimnei 17952 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.)

Theoremflimelbas 17953 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.)

Theoremflimfil 17954 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremflimtopon 17955 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremelflim 17956 The predicate "is a limit point of a filter." (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
TopOn

Theoremflimss2 17957 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.)
TopOn

Theoremflimss1 17958 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.)
TopOn

Theoremneiflim 17959 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.)
TopOn

Theoremflimopn 17960* 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.)
TopOn

Theoremfbflim 17961* 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.)
TopOn

Theoremfbflim2 17962* A condition for a filter base to converge to a point . Use neighborhoods instead of open neighborhoods. Compare fbflim 17961. (Contributed by FL, 4-Jul-2011.) (Revised by Stefan O'Rear, 6-Aug-2015.)
TopOn

Theoremflimclsi 17963 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.)

Theoremhausflimlem 17964 If and are both limits of the same filter, then all neighborhoods of and intersect. (Contributed by Mario Carneiro, 21-Sep-2015.)

Theoremhausflimi 17965* One direction of hausflim 17966. A filter in a Hausdorf space has at most one limit. (Contributed by FL, 14-Nov-2010.) (Revised by Mario Carneiro, 21-Sep-2015.)

Theoremhausflim 17966* 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.)

Theoremflimcf 17967* 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.)
TopOn TopOn

Theoremflimrest 17968 The set of limit points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
TopOn t t

Theoremflimclslem 17969 Lemma for flimcls 17970. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
TopOn

Theoremflimcls 17970* Closure in terms of filter convergence. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
TopOn

Theoremflimsncls 17971 If is a limit point of the filter , then all the points which specialize (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.)

Theoremhauspwpwf1 17972* Lemma for hauspwpwdom 17973. 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.)

Theoremhauspwpwdom 17973 If is a Hausdorff space, then the cardinality of the closure of a set is bounded by the double powerset of . In particular, a Hausdorff space with a dense subset has cardinality at most , and a separable Hausdorff space has cardinality at most . (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremflffval 17974* 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.)
TopOn

Theoremflfval 17975 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.)
TopOn

Theoremflfnei 17976* 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.)
TopOn

Theoremflfneii 17977* 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.)

Theoremisflf 17978* 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.)
TopOn

Theoremflfelbas 17979 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.)
TopOn

Theoremflffbas 17980* 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.)
TopOn

Theoremflftg 17981* Limit points of a function can be defined using topological bases. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopOn

Theoremhausflf 17982* 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.)

Theoremhausflf2 17983 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.)

Theoremcnpflfi 17984 Forward direction of cnpflf 17986. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Theoremcnpflf2 17985 is continous at point iff a limit of when tends to is . Proposition 9 of [BourbakiTop1] p. TG I.50. (Contributed by FL, 29-May-2011.) (Revised by Mario Carneiro, 9-Apr-2015.)
TopOn TopOn

Theoremcnpflf 17986* 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.)
TopOn TopOn

Theoremcnflf 17987* A function is continuous iff it respects filter limits. (Contributed by Jeff Hankins, 6-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
TopOn TopOn

Theoremcnflf2 17988* A function is continuous iff it respects filter limits. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
TopOn TopOn

Theoremflfcnp 17989 A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.)
TopOn

Theoremlmflf 17990 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.)
TopOn

Theoremtxflf 17991* Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopOn       TopOn

Theoremflfcnp2 17992* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopOn       TopOn

Theoremfclsval 17993* The set of all cluster points of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremisfcls 17994* A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclsfil 17995 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclstop 17996 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfclstopon 17997 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremisfcls2 17998* A cluster point of a filter. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremfclsopn 17999* Write the cluster point condition in terms of open sets. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremfclsopni 18000 An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

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