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Type | Label | Description |
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Statement | ||
Theorem | elfilss 23601* | An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
β’ ((πΉ β (Filβπ) β§ π΄ β π) β (π΄ β πΉ β βπ‘ β πΉ π‘ β π΄)) | ||
Theorem | filfinnfr 23602 | No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ ((πΉ β (Filβπ) β§ π β πΉ β§ π β Fin) β β© πΉ β β ) | ||
Theorem | fgcl 23603 | A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ (πΉ β (fBasβπ) β (πfilGenπΉ) β (Filβπ)) | ||
Theorem | fgabs 23604 | Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ ((πΉ β (fBasβπ) β§ π β π) β (πfilGen(πfilGenπΉ)) = (πfilGenπΉ)) | ||
Theorem | neifil 23605 | The neighborhoods of a nonempty set is a filter. Example 2 of [BourbakiTop1] p. I.36. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
β’ ((π½ β (TopOnβπ) β§ π β π β§ π β β ) β ((neiβπ½)βπ) β (Filβπ)) | ||
Theorem | filunibas 23606 | Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
β’ (πΉ β (Filβπ) β βͺ πΉ = π) | ||
Theorem | filunirn 23607 | Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
β’ (πΉ β βͺ ran Fil β πΉ β (Filββͺ πΉ)) | ||
Theorem | filconn 23608 | A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ (πΉ β (Filβπ) β (πΉ βͺ {β }) β Conn) | ||
Theorem | fbasrn 23609* | 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.) |
β’ πΆ = ran (π₯ β π΅ β¦ (πΉ β π₯)) β β’ ((π΅ β (fBasβπ) β§ πΉ:πβΆπ β§ π β π) β πΆ β (fBasβπ)) | ||
Theorem | filuni 23610* | 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.) |
β’ ((πΉ β (Filβπ) β§ πΉ β β β§ βπ β πΉ βπ β πΉ (π βͺ π) β πΉ) β βͺ πΉ β (Filβπ)) | ||
Theorem | trfil1 23611 | Conditions for the trace of a filter πΏ to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ ((πΏ β (Filβπ) β§ π΄ β π) β π΄ = βͺ (πΏ βΎt π΄)) | ||
Theorem | trfil2 23612* | Conditions for the trace of a filter πΏ to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ ((πΏ β (Filβπ) β§ π΄ β π) β ((πΏ βΎt π΄) β (Filβπ΄) β βπ£ β πΏ (π£ β© π΄) β β )) | ||
Theorem | trfil3 23613 | Conditions for the trace of a filter πΏ to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
β’ ((πΏ β (Filβπ) β§ π΄ β π) β ((πΏ βΎt π΄) β (Filβπ΄) β Β¬ (π β π΄) β πΏ)) | ||
Theorem | trfilss 23614 | If π΄ is a member of the filter, then the filter truncated to π΄ is a subset of the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ ((πΉ β (Filβπ) β§ π΄ β πΉ) β (πΉ βΎt π΄) β πΉ) | ||
Theorem | fgtr 23615 | If π΄ is a member of the filter, then truncating πΉ to π΄ and regenerating the behavior outside π΄ using filGen recovers the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ ((πΉ β (Filβπ) β§ π΄ β πΉ) β (πfilGen(πΉ βΎt π΄)) = πΉ) | ||
Theorem | trfg 23616 | The trace operation and the filGen operation are inverses to one another in some sense, with filGen growing the base set and βΎt shrinking it. See fgtr 23615 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ ((πΉ β (Filβπ΄) β§ π΄ β π β§ π β π) β ((πfilGenπΉ) βΎt π΄) = πΉ) | ||
Theorem | trnei 23617 | The trace, over a set π΄, of the filter of the neighborhoods of a point π is a filter iff π belongs to the closure of π΄. (This is trfil2 23612 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ ((π½ β (TopOnβπ) β§ π΄ β π β§ π β π) β (π β ((clsβπ½)βπ΄) β (((neiβπ½)β{π}) βΎt π΄) β (Filβπ΄))) | ||
Theorem | cfinfil 23618* | Relative complements of the finite parts of an infinite set is a filter. When π΄ = β 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.) |
β’ ((π β π β§ π΄ β π β§ Β¬ π΄ β Fin) β {π₯ β π« π β£ (π΄ β π₯) β Fin} β (Filβπ)) | ||
Theorem | csdfil 23619* | 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.) |
β’ ((π β dom card β§ Ο βΌ π) β {π₯ β π« π β£ (π β π₯) βΊ π} β (Filβπ)) | ||
Theorem | supfil 23620* | 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.) |
β’ ((π΄ β π β§ π΅ β π΄ β§ π΅ β β ) β {π₯ β π« π΄ β£ π΅ β π₯} β (Filβπ΄)) | ||
Theorem | zfbas 23621 | The set of upper sets of integers is a filter base on β€, which corresponds to convergence of sequences on β€. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ ran β€β₯ β (fBasββ€) | ||
Theorem | uzrest 23622 | The restriction of the set of upper sets of integers to an upper set of integers is the set of upper sets of integers based at a point above the cutoff. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (β€β₯βπ) β β’ (π β β€ β (ran β€β₯ βΎt π) = (β€β₯ β π)) | ||
Theorem | uzfbas 23623 | The set of upper sets of integers based at a point in a fixed upper integer set like β is a filter base on β, which corresponds to convergence of sequences on β. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (β€β₯βπ) β β’ (π β β€ β (β€β₯ β π) β (fBasβπ)) | ||
Syntax | cufil 23624 | Extend class notation with the ultrafilters-on-a-set function. |
class UFil | ||
Syntax | cufl 23625 | Extend class notation with the ultrafilter lemma. |
class UFL | ||
Definition | df-ufil 23626* | 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.) |
β’ UFil = (π β V β¦ {π β (Filβπ) β£ βπ₯ β π« π(π₯ β π β¨ (π β π₯) β π)}) | ||
Definition | df-ufl 23627* | Define the class of base sets for which the ultrafilter lemma filssufil 23637 holds. (Contributed by Mario Carneiro, 26-Aug-2015.) |
β’ UFL = {π₯ β£ βπ β (Filβπ₯)βπ β (UFilβπ₯)π β π} | ||
Theorem | isufil 23628* | The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.) |
β’ (πΉ β (UFilβπ) β (πΉ β (Filβπ) β§ βπ₯ β π« π(π₯ β πΉ β¨ (π β π₯) β πΉ))) | ||
Theorem | ufilfil 23629 | An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.) |
β’ (πΉ β (UFilβπ) β πΉ β (Filβπ)) | ||
Theorem | ufilss 23630 | 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.) |
β’ ((πΉ β (UFilβπ) β§ π β π) β (π β πΉ β¨ (π β π) β πΉ)) | ||
Theorem | ufilb 23631 | 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.) |
β’ ((πΉ β (UFilβπ) β§ π β π) β (Β¬ π β πΉ β (π β π) β πΉ)) | ||
Theorem | ufilmax 23632 | 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.) |
β’ ((πΉ β (UFilβπ) β§ πΊ β (Filβπ) β§ πΉ β πΊ) β πΉ = πΊ) | ||
Theorem | isufil2 23633* | The maximal property of an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ (πΉ β (UFilβπ) β (πΉ β (Filβπ) β§ βπ β (Filβπ)(πΉ β π β πΉ = π))) | ||
Theorem | ufprim 23634 | An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.) |
β’ ((πΉ β (UFilβπ) β§ π΄ β π β§ π΅ β π) β ((π΄ β πΉ β¨ π΅ β πΉ) β (π΄ βͺ π΅) β πΉ)) | ||
Theorem | trufil 23635 | Conditions for the trace of an ultrafilter πΏ to be an ultrafilter. (Contributed by Mario Carneiro, 27-Aug-2015.) |
β’ ((πΏ β (UFilβπ) β§ π΄ β π) β ((πΏ βΎt π΄) β (UFilβπ΄) β π΄ β πΏ)) | ||
Theorem | filssufilg 23636* | A filter is contained in some ultrafilter. This version of filssufil 23637 contains the choice as a hypothesis (in the assumption that π« π« π is well-orderable). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ ((πΉ β (Filβπ) β§ π« π« π β dom card) β βπ β (UFilβπ)πΉ β π) | ||
Theorem | filssufil 23637* | A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 10468.) (Contributed by Jeff Hankins, 2-Dec-2009.) (Revised by Stefan O'Rear, 29-Jul-2015.) |
β’ (πΉ β (Filβπ) β βπ β (UFilβπ)πΉ β π) | ||
Theorem | isufl 23638* | Define the (strong) ultrafilter lemma, parameterized over base sets. A set π satisfies the ultrafilter lemma if every filter on π is a subset of some ultrafilter. (Contributed by Mario Carneiro, 26-Aug-2015.) |
β’ (π β π β (π β UFL β βπ β (Filβπ)βπ β (UFilβπ)π β π)) | ||
Theorem | ufli 23639* | Property of a set that satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.) |
β’ ((π β UFL β§ πΉ β (Filβπ)) β βπ β (UFilβπ)πΉ β π) | ||
Theorem | numufl 23640 | Consequence of filssufilg 23636: a set whose double powerset is well-orderable satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.) |
β’ (π« π« π β dom card β π β UFL) | ||
Theorem | fiufl 23641 | A finite set satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.) |
β’ (π β Fin β π β UFL) | ||
Theorem | acufl 23642 | The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.) |
β’ (CHOICE β UFL = V) | ||
Theorem | ssufl 23643 | If π is a subset of π and filters extend to ultrafilters in π, then they still do in π. (Contributed by Mario Carneiro, 26-Aug-2015.) |
β’ ((π β UFL β§ π β π) β π β UFL) | ||
Theorem | ufileu 23644* | 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.) |
β’ (πΉ β (Filβπ) β (πΉ β (UFilβπ) β β!π β (UFilβπ)πΉ β π)) | ||
Theorem | filufint 23645* | 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.) |
β’ (πΉ β (Filβπ) β β© {π β (UFilβπ) β£ πΉ β π} = πΉ) | ||
Theorem | uffix 23646* | Lemma for fixufil 23647 and uffixfr 23648. (Contributed by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ ((π β π β§ π΄ β π) β ({{π΄}} β (fBasβπ) β§ {π₯ β π« π β£ π΄ β π₯} = (πfilGen{{π΄}}))) | ||
Theorem | fixufil 23647* | 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.) |
β’ ((π β π β§ π΄ β π) β {π₯ β π« π β£ π΄ β π₯} β (UFilβπ)) | ||
Theorem | uffixfr 23648* | An ultrafilter is either fixed or free. A fixed ultrafilter is called principal (generated by a single element π΄), 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.) |
β’ (πΉ β (UFilβπ) β (π΄ β β© πΉ β πΉ = {π₯ β π« π β£ π΄ β π₯})) | ||
Theorem | uffix2 23649* | A classification of fixed ultrafilters. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ (πΉ β (UFilβπ) β (β© πΉ β β β βπ₯ β π πΉ = {π¦ β π« π β£ π₯ β π¦})) | ||
Theorem | uffixsn 23650 | 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.) |
β’ ((πΉ β (UFilβπ) β§ π΄ β β© πΉ) β {π΄} β πΉ) | ||
Theorem | ufildom1 23651 | 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.) |
β’ (πΉ β (UFilβπ) β β© πΉ βΌ 1o) | ||
Theorem | uffinfix 23652* | An ultrafilter containing a finite element is fixed. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ ((πΉ β (UFilβπ) β§ π β πΉ β§ π β Fin) β βπ₯ β π πΉ = {π¦ β π« π β£ π₯ β π¦}) | ||
Theorem | cfinufil 23653* | An ultrafilter is free iff it contains the FrΓ©chet filter cfinfil 23618 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
β’ (πΉ β (UFilβπ) β (β© πΉ = β β {π₯ β π« π β£ (π β π₯) β Fin} β πΉ)) | ||
Theorem | ufinffr 23654* | An infinite subset is contained in a free ultrafilter. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Mario Carneiro, 4-Dec-2013.) |
β’ ((π β π΅ β§ π΄ β π β§ Ο βΌ π΄) β βπ β (UFilβπ)(π΄ β π β§ β© π = β )) | ||
Theorem | ufilen 23655* | 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.) |
β’ (Ο βΌ π β βπ β (UFilβπ)βπ₯ β π π₯ β π) | ||
Theorem | ufildr 23656 | An ultrafilter gives rise to a connected door topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.) |
β’ π½ = (πΉ βͺ {β }) β β’ (πΉ β (UFilβπ) β (π½ βͺ (Clsdβπ½)) = π« π) | ||
Theorem | fin1aufil 23657 | 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) π, the set of infinite subsets of π is a free ultrafilter on π. (Contributed by Mario Carneiro, 20-May-2015.) |
β’ πΉ = (π« π β Fin) β β’ (π β (FinIa β Fin) β (πΉ β (UFilβπ) β§ β© πΉ = β )) | ||
Syntax | cfm 23658 | Extend class definition to include the neighborhood filter mapping function. |
class FilMap | ||
Syntax | cflim 23659 | Extend class notation with a function returning the limit of a filter. |
class fLim | ||
Syntax | cflf 23660 | Extend class definition to include the function for filter-based function limits. |
class fLimf | ||
Syntax | cfcls 23661 | Extend class definition to include the cluster point function on filters. |
class fClus | ||
Syntax | cfcf 23662 | Extend class definition to include the function for cluster points of a function. |
class fClusf | ||
Definition | df-fm 23663* | 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.) |
β’ FilMap = (π₯ β V, π β V β¦ (π¦ β (fBasβdom π) β¦ (π₯filGenran (π‘ β π¦ β¦ (π β π‘))))) | ||
Definition | df-flim 23664* | Define a function (indexed by a topology π) whose value is the limits of a filter π. (Contributed by Jeff Hankins, 4-Sep-2009.) |
β’ fLim = (π β Top, π β βͺ ran Fil β¦ {π₯ β βͺ π β£ (((neiβπ)β{π₯}) β π β§ π β π« βͺ π)}) | ||
Definition | df-flf 23665* | Define a function that gives the limits of a function π in the filter sense. (Contributed by Jeff Hankins, 14-Oct-2009.) |
β’ fLimf = (π₯ β Top, π¦ β βͺ ran Fil β¦ (π β (βͺ π₯ βm βͺ π¦) β¦ (π₯ fLim ((βͺ π₯ FilMap π)βπ¦)))) | ||
Definition | df-fcls 23666* | Define a function that takes a filter in a topology to its set of cluster points. (Contributed by Jeff Hankins, 10-Nov-2009.) |
β’ fClus = (π β Top, π β βͺ ran Fil β¦ if(βͺ π = βͺ π, β© π₯ β π ((clsβπ)βπ₯), β )) | ||
Definition | df-fcf 23667* | Define a function that gives the cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) |
β’ fClusf = (π β Top, π β βͺ ran Fil β¦ (π β (βͺ π βm βͺ π) β¦ (π fClus ((βͺ π FilMap π)βπ)))) | ||
Theorem | fmval 23668* | 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 positive integer 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.) |
β’ ((π β π΄ β§ π΅ β (fBasβπ) β§ πΉ:πβΆπ) β ((π FilMap πΉ)βπ΅) = (πfilGenran (π¦ β π΅ β¦ (πΉ β π¦)))) | ||
Theorem | fmfil 23669 | A mapping filter is a filter. (Contributed by Jeff Hankins, 18-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
β’ ((π β π΄ β§ π΅ β (fBasβπ) β§ πΉ:πβΆπ) β ((π FilMap πΉ)βπ΅) β (Filβπ)) | ||
Theorem | fmf 23670 | Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.) |
β’ ((π β π΄ β§ π β π΅ β§ πΉ:πβΆπ) β (π FilMap πΉ):(fBasβπ)βΆ(Filβπ)) | ||
Theorem | fmss 23671 | A finer filter produces a finer image filter. (Contributed by Jeff Hankins, 16-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
β’ (((π β π΄ β§ π΅ β (fBasβπ) β§ πΆ β (fBasβπ)) β§ (πΉ:πβΆπ β§ π΅ β πΆ)) β ((π FilMap πΉ)βπ΅) β ((π FilMap πΉ)βπΆ)) | ||
Theorem | elfm 23672* | An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
β’ ((π β πΆ β§ π΅ β (fBasβπ) β§ πΉ:πβΆπ) β (π΄ β ((π FilMap πΉ)βπ΅) β (π΄ β π β§ βπ₯ β π΅ (πΉ β π₯) β π΄))) | ||
Theorem | elfm2 23673* | An element of a mapping filter. (Contributed by Jeff Hankins, 26-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
β’ πΏ = (πfilGenπ΅) β β’ ((π β πΆ β§ π΅ β (fBasβπ) β§ πΉ:πβΆπ) β (π΄ β ((π FilMap πΉ)βπ΅) β (π΄ β π β§ βπ₯ β πΏ (πΉ β π₯) β π΄))) | ||
Theorem | fmfg 23674 | 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.) |
β’ πΏ = (πfilGenπ΅) β β’ ((π β πΆ β§ π΅ β (fBasβπ) β§ πΉ:πβΆπ) β ((π FilMap πΉ)βπ΅) = ((π FilMap πΉ)βπΏ)) | ||
Theorem | elfm3 23675* | An alternate formulation of elementhood in a mapping filter that requires πΉ to be onto. (Contributed by Jeff Hankins, 1-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
β’ πΏ = (πfilGenπ΅) β β’ ((π΅ β (fBasβπ) β§ πΉ:πβontoβπ) β (π΄ β ((π FilMap πΉ)βπ΅) β βπ₯ β πΏ π΄ = (πΉ β π₯))) | ||
Theorem | imaelfm 23676 | 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.) |
β’ πΏ = (πfilGenπ΅) β β’ (((π β π΄ β§ π΅ β (fBasβπ) β§ πΉ:πβΆπ) β§ π β πΏ) β (πΉ β π) β ((π FilMap πΉ)βπ΅)) | ||
Theorem | rnelfmlem 23677* | Lemma for rnelfm 23678. (Contributed by Jeff Hankins, 14-Nov-2009.) |
β’ (((π β π΄ β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β§ ran πΉ β πΏ) β ran (π₯ β πΏ β¦ (β‘πΉ β π₯)) β (fBasβπ)) | ||
Theorem | rnelfm 23678 | 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.) |
β’ ((π β π΄ β§ πΏ β (Filβπ) β§ πΉ:πβΆπ) β (πΏ β ran (π FilMap πΉ) β ran πΉ β πΏ)) | ||
Theorem | fmfnfmlem1 23679* | Lemma for fmfnfm 23683. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
β’ (π β π΅ β (fBasβπ)) & β’ (π β πΏ β (Filβπ)) & β’ (π β πΉ:πβΆπ) & β’ (π β ((π FilMap πΉ)βπ΅) β πΏ) β β’ (π β (π β (fiβπ΅) β ((πΉ β π ) β π‘ β (π‘ β π β π‘ β πΏ)))) | ||
Theorem | fmfnfmlem2 23680* | Lemma for fmfnfm 23683. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
β’ (π β π΅ β (fBasβπ)) & β’ (π β πΏ β (Filβπ)) & β’ (π β πΉ:πβΆπ) & β’ (π β ((π FilMap πΉ)βπ΅) β πΏ) β β’ (π β (βπ₯ β πΏ π = (β‘πΉ β π₯) β ((πΉ β π ) β π‘ β (π‘ β π β π‘ β πΏ)))) | ||
Theorem | fmfnfmlem3 23681* | Lemma for fmfnfm 23683. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
β’ (π β π΅ β (fBasβπ)) & β’ (π β πΏ β (Filβπ)) & β’ (π β πΉ:πβΆπ) & β’ (π β ((π FilMap πΉ)βπ΅) β πΏ) β β’ (π β (fiβran (π₯ β πΏ β¦ (β‘πΉ β π₯))) = ran (π₯ β πΏ β¦ (β‘πΉ β π₯))) | ||
Theorem | fmfnfmlem4 23682* | Lemma for fmfnfm 23683. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
β’ (π β π΅ β (fBasβπ)) & β’ (π β πΏ β (Filβπ)) & β’ (π β πΉ:πβΆπ) & β’ (π β ((π FilMap πΉ)βπ΅) β πΏ) β β’ (π β (π‘ β πΏ β (π‘ β π β§ βπ β (fiβ(π΅ βͺ ran (π₯ β πΏ β¦ (β‘πΉ β π₯))))(πΉ β π ) β π‘))) | ||
Theorem | fmfnfm 23683* | 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.) |
β’ (π β π΅ β (fBasβπ)) & β’ (π β πΏ β (Filβπ)) & β’ (π β πΉ:πβΆπ) & β’ (π β ((π FilMap πΉ)βπ΅) β πΏ) β β’ (π β βπ β (Filβπ)(π΅ β π β§ πΏ = ((π FilMap πΉ)βπ))) | ||
Theorem | fmufil 23684 | An image filter of an ultrafilter is an ultrafilter. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
β’ ((π β π΄ β§ πΏ β (UFilβπ) β§ πΉ:πβΆπ) β ((π FilMap πΉ)βπΏ) β (UFilβπ)) | ||
Theorem | fmid 23685 | The filter map applied to the identity. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Mario Carneiro, 27-Aug-2015.) |
β’ (πΉ β (Filβπ) β ((π FilMap ( I βΎ π))βπΉ) = πΉ) | ||
Theorem | fmco 23686 | Composition of image filters. (Contributed by Mario Carneiro, 27-Aug-2015.) |
β’ (((π β π β§ π β π β§ π΅ β (fBasβπ)) β§ (πΉ:πβΆπ β§ πΊ:πβΆπ)) β ((π FilMap (πΉ β πΊ))βπ΅) = ((π FilMap πΉ)β((π FilMap πΊ)βπ΅))) | ||
Theorem | ufldom 23687 | 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.) |
β’ ((π β UFL β§ π βΌ π) β π β UFL) | ||
Theorem | flimval 23688* | The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
β’ π = βͺ π½ β β’ ((π½ β Top β§ πΉ β βͺ ran Fil) β (π½ fLim πΉ) = {π₯ β π β£ (((neiβπ½)β{π₯}) β πΉ β§ πΉ β π« π)}) | ||
Theorem | elflim2 23689 | The predicate "is a limit point of a filter." (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
β’ π = βͺ π½ β β’ (π΄ β (π½ fLim πΉ) β ((π½ β Top β§ πΉ β βͺ ran Fil β§ πΉ β π« π) β§ (π΄ β π β§ ((neiβπ½)β{π΄}) β πΉ))) | ||
Theorem | flimtop 23690 | Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
β’ (π΄ β (π½ fLim πΉ) β π½ β Top) | ||
Theorem | flimneiss 23691 | A filter contains the neighborhood filter as a subfilter. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
β’ (π΄ β (π½ fLim πΉ) β ((neiβπ½)β{π΄}) β πΉ) | ||
Theorem | flimnei 23692 | 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.) |
β’ ((π΄ β (π½ fLim πΉ) β§ π β ((neiβπ½)β{π΄})) β π β πΉ) | ||
Theorem | flimelbas 23693 | 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.) |
β’ π = βͺ π½ β β’ (π΄ β (π½ fLim πΉ) β π΄ β π) | ||
Theorem | flimfil 23694 | Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
β’ π = βͺ π½ β β’ (π΄ β (π½ fLim πΉ) β πΉ β (Filβπ)) | ||
Theorem | flimtopon 23695 | Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.) |
β’ (π΄ β (π½ fLim πΉ) β (π½ β (TopOnβπ) β πΉ β (Filβπ))) | ||
Theorem | elflim 23696 | The predicate "is a limit point of a filter." (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.) |
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β (π΄ β (π½ fLim πΉ) β (π΄ β π β§ ((neiβπ½)β{π΄}) β πΉ))) | ||
Theorem | flimss2 23697 | 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.) |
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ) β§ πΊ β πΉ) β (π½ fLim πΊ) β (π½ fLim πΉ)) | ||
Theorem | flimss1 23698 | 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.) |
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ) β§ π½ β πΎ) β (πΎ fLim πΉ) β (π½ fLim πΉ)) | ||
Theorem | neiflim 23699 | 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.) |
β’ ((π½ β (TopOnβπ) β§ π΄ β π) β π΄ β (π½ fLim ((neiβπ½)β{π΄}))) | ||
Theorem | flimopn 23700* | 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.) |
β’ ((π½ β (TopOnβπ) β§ πΉ β (Filβπ)) β (π΄ β (π½ fLim πΉ) β (π΄ β π β§ βπ₯ β π½ (π΄ β π₯ β π₯ β πΉ)))) |
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