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

Theoremfbunfip 17901* A helpful lemma for showing that certain sets generate filters. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgval 17902* The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremelfg 17903* A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremssfg 17904 A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgss 17905 A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgss2 17906* A condition for a filter to be finer than another involving their filter bases. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgfil 17907 A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremelfilss 17908* An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilfinnfr 17909 No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgcl 17910 A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgabs 17911 Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremneifil 17912 The neighborhoods of a non-empty set is a filter. Example 2 of [BourbakiTop1] p. I.36. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
TopOn

Theoremfilunibas 17913 Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilunirn 17914 Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilcon 17915 A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfbasrn 17916* Given a filter on a domain, produce a filter on the range. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremfiluni 17917* The union of a nonempty set of filters with a common base and closed under pairwise union is a filter. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremtrfil1 17918 Conditions for the trace of a filter to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
t

Theoremtrfil2 17919* Conditions for the trace of a filter to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
t

Theoremtrfil3 17920 Conditions for the trace of a filter to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
t

Theoremtrfilss 17921 If is a member of the filter, then the filter truncated to is a subset of the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
t

Theoremfgtr 17922 If is a member of the filter, then truncating to and regenerating the behavior outside using recovers the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
t

Theoremtrfg 17923 The trace operation and the operation are inverses to one another in some sense, with growing the base set and ↾t shrinking it. See fgtr 17922 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)
t

Theoremtrnei 17924 The trace, over a set , of the filter of the neighborhoods of a point is a filter iff belongs to the closure of . (This is trfil2 17919 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
TopOn t

Theoremcfinfil 17925* Relative complements of the finite parts of an infinite set is a filter. When 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.)

Theoremcsdfil 17926* 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.)

Theoremsupfil 17927* 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.)

Theoremzfbas 17928 The set of upper integer sets is a filter base on , which corresponds to convergence of sequences on . (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremuzrest 17929 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.)
t

Theoremuzfbas 17930 The set of upper integer sets based at a point in a fixed upper integer set like is a filter base on , which corresponds to convergence of sequences on . (Contributed by Mario Carneiro, 13-Oct-2015.)

11.2.3  Ultrafilters

Syntaxcufil 17931 Extend class notation with the ultrafilters-on-a-set function.

Syntaxcufl 17932 Extend class notation with the ultrafilter lemma.
UFL

Definitiondf-ufil 17933* 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.)

Definitiondf-ufl 17934* Define the class of base sets for which the ultrafilter lemma filssufil 17944 holds. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremisufil 17935* The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Theoremufilfil 17936 An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Theoremufilss 17937 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.)

Theoremufilb 17938 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.)

Theoremufilmax 17939 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.)

Theoremisufil2 17940* The maximal property of an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremufprim 17941 An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.)

Theoremtrufil 17942 Conditions for the trace of an ultrafilter to be an ultrafilter. (Contributed by Mario Carneiro, 27-Aug-2015.)
t

Theoremfilssufilg 17943* A filter is contained in some ultrafilter. This version of filssufil 17944 contains the choice as a hypothesis (in the assumption that is well-orderable). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfilssufil 17944* A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 8350.) (Contributed by Jeff Hankins, 2-Dec-2009.) (Revised by Stefan O'Rear, 29-Jul-2015.)

Theoremisufl 17945* Define the (strong) ultrafilter lemma, parameterized over base sets. A set satisfies the ultrafilter lemma if every filter on is a subset of some ultrafilter. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremufli 17946* Property of a set that satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremnumufl 17947 Consequence of filssufilg 17943: a set whose double powerset is well-orderable satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremfiufl 17948 A finite set satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremacufl 17949 The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
CHOICE UFL

Theoremssufl 17950 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 17951* 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 17952* 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 17953* Lemma for fixufil 17954 and uffixfr 17955. (Contributed by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfixufil 17954* 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 17955* 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 17956* A classification of fixed ultrafilters. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremuffixsn 17957 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 17958 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 17959* An ultrafilter containing a finite element is fixed. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremcfinufil 17960* An ultrafilter is free iff it contains the Fréchet filter cfinfil 17925 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremufinffr 17961* An infinite subset is contained in a free ultrafilter. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Mario Carneiro, 4-Dec-2013.)

Theoremufilen 17962* 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 17963 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 17964 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 17965 Extend class definition to include the neighborhood filter mapping function.

Syntaxcflim 17966 Extend class notation with a function returning the limit of a filter.

Syntaxcflf 17967 Extend class definition to include the function for filter-based function limits.

Syntaxcfcls 17968 Extend class definition to include the cluster point function on filters.

Syntaxcfcf 17969 Extend class definition to include the function for cluster points of a function.

Definitiondf-fm 17970* 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 17971* Define a function (indexed by a topology ) whose value is the limits of a filter . (Contributed by Jeff Hankins, 4-Sep-2009.)

Definitiondf-flf 17972* Define a function that gives the limits of a function in the filter sense. (Contributed by Jeff Hankins, 14-Oct-2009.)

Definitiondf-fcls 17973* 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 17974* Define a function that gives the cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.)

Theoremfmval 17975* 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 17976 A mapping filter is a filter. (Contributed by Jeff Hankins, 18-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremfmf 17977 Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.)

Theoremfmss 17978 A finer filter produces a finer image filter. (Contributed by Jeff Hankins, 16-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremelfm 17979* An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremelfm2 17980* An element of a mapping filter. (Contributed by Jeff Hankins, 26-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremfmfg 17981 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 17982* 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 17983 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 17984* Lemma for rnelfm 17985. (Contributed by Jeff Hankins, 14-Nov-2009.)

Theoremrnelfm 17985 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 17986* Lemma for fmfnfm 17990. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfmlem2 17987* Lemma for fmfnfm 17990. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfmlem3 17988* Lemma for fmfnfm 17990. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfmlem4 17989* Lemma for fmfnfm 17990. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfm 17990* 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 17991 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 17992 The filter map applied to the identity. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Mario Carneiro, 27-Aug-2015.)

Theoremfmco 17993 Composition of image filters. (Contributed by Mario Carneiro, 27-Aug-2015.)

Theoremufldom 17994 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 17995* The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremelflim2 17996 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 17997 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Theoremflimneiss 17998 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 17999 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 18000 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.)

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