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Theorem List for Metamath Proof Explorer - 17401-17500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfgss2 17401* 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 17402 A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremelfilss 17403* An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilfinnfr 17404 No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgcl 17405 A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgabs 17406 Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremneifil 17407 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 17408 Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilunirn 17409 Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilcon 17410 A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfbasrn 17411* 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 17412* 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 17413 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 17414* 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 17415 Conditions for the trace of a filter to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
t

Theoremtrfilss 17416 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 17417 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 17418 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 17417 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)
t

Theoremtrnei 17419 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 17414 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
TopOn t

Theoremcfinfil 17420* 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 17421* 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 17422* 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 17423 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 17424 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 17425 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 17426 Extend class notation with the ultrafilters-on-a-set function.

Syntaxcufl 17427 Extend class notation with the ultrafilter lemma.
UFL

Definitiondf-ufil 17428* 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 17429* Define the class of base sets for which the ultrafilter lemma filssufil 17439 holds. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremisufil 17430* The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Theoremufilfil 17431 An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Theoremufilss 17432 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 17433 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 17434 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 17435* The maximal property of an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremufprim 17436 An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.)

Theoremtrufil 17437 Conditions for the trace of an ultrafilter to be an ultrafilter. (Contributed by Mario Carneiro, 27-Aug-2015.)
t

Theoremfilssufilg 17438* A filter is contained in some ultrafilter. This version of filssufil 17439 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 17439* A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 7981.) (Contributed by Jeff Hankins, 2-Dec-2009.) (Revised by Stefan O'Rear, 29-Jul-2015.)

Theoremisufl 17440* 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 17441* Property of a set that satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremnumufl 17442 Consequence of filssufilg 17438: a set whose double powerset is well-orderable satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremfiufl 17443 A finite set satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremacufl 17444 The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
CHOICE UFL

Theoremssufl 17445 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 17446* 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 17447* 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 17448* Lemma for fixufil 17449 and uffixfr 17450. (Contributed by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfixufil 17449* 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 17450* 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 17451* A classification of fixed ultrafilters. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremuffixsn 17452 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 17453 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 17454* An ultrafilter containing a finite element is fixed. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremcfinufil 17455* An ultrafilter is free iff it contains the Fréchet filter cfinfil 17420 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremufinffr 17456* An infinite subset is contained in a free ultrafilter. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Mario Carneiro, 4-Dec-2013.)

Theoremufilen 17457* 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 17458 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 17459 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 17460 Extend class definition to include the neighborhood filter mapping function.

Syntaxcflim 17461 Extend class notation with a function returning the limit of a filter.

Syntaxcflf 17462 Extend class definition to include the function for filter-based function limits.

Syntaxcfcls 17463 Extend class definition to include the cluster point function on filters.

Syntaxcfcf 17464 Extend class definition to include the function for cluster points of a function.

Definitiondf-fm 17465* 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 17466* Define a function (indexed by a topology ) whose value is the limits of a filter . (Contributed by Jeff Hankins, 4-Sep-2009.)

Definitiondf-flf 17467* Define a function that gives the limits of a function in the filter sense. (Contributed by Jeff Hankins, 14-Oct-2009.)

Definitiondf-fcls 17468* 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 17469* Define a function that gives the cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.)

Theoremfmval 17470* 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 17471 A mapping filter is a filter. (Contributed by Jeff Hankins, 18-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremfmf 17472 Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.)

Theoremfmss 17473 A finer filter produces a finer image filter. (Contributed by Jeff Hankins, 16-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremelfm 17474* An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremelfm2 17475* An element of a mapping filter. (Contributed by Jeff Hankins, 26-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremfmfg 17476 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 17477* 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 17478 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 17479* Lemma for rnelfm 17480. (Contributed by Jeff Hankins, 14-Nov-2009.)

Theoremrnelfm 17480 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 17481* Lemma for fmfnfm 17485. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfmlem2 17482* Lemma for fmfnfm 17485. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfmlem3 17483* Lemma for fmfnfm 17485. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfmlem4 17484* Lemma for fmfnfm 17485. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Theoremfmfnfm 17485* 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 17486 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 17487 The filter map applied to the identity. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Mario Carneiro, 27-Aug-2015.)

Theoremfmco 17488 Composition of image filters. (Contributed by Mario Carneiro, 27-Aug-2015.)

Theoremufldom 17489 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 17490* The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremelflim2 17491 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 17492 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Theoremflimneiss 17493 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 17494 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 17495 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 17496 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremflimtopon 17497 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn

Theoremelflim 17498 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 17499 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 17500 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

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