P. 211 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isufil2 21001*	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 ) ) )
Theorem	ufprim 21002	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 ) )
Theorem	trufil 21003	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 ) -> ( ( L ↾t A ) e. ( UFil ` A ) <-> A e. L ) )
Theorem	filssufilg 21004*	A filter is contained in some ultrafilter. This version of filssufil 21005 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 )
Theorem	filssufil 21005*	A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 8918.) (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 )
Theorem	isufl 21006*	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 ) )
Theorem	ufli 21007*	Property of a set that satisfies 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 )
Theorem	numufl 21008	Consequence of filssufilg 21004: a set whose double powerset is well-orderable satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( ~P ~P X e. dom card -> X e. UFL )
Theorem	fiufl 21009	A finite set satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( X e. Fin -> X e. UFL )
Theorem	acufl 21010	The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- (CHOICE -> UFL = _V )
Theorem	ssufl 21011	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 )
Theorem	ufileu 21012*	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 ) )
Theorem	filufint 21013*	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 )
Theorem	uffix 21014*	Lemma for fixufil 21015 and uffixfr 21016. (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 } } ) ) )
Theorem	fixufil 21015*	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 ) )
Theorem	uffixfr 21016*	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 } ) )
Theorem	uffix2 21017*	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 } ) )
Theorem	uffixsn 21018	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 )
Theorem	ufildom1 21019	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 )
Theorem	uffinfix 21020*	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 } )
Theorem	cfinufil 21021*	An ultrafilter is free iff it contains the Fréchet filter cfinfil 20986 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 ) )
Theorem	ufinffr 21022*	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 = (/) ) )
Theorem	ufilen 21023*	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 )
Theorem	ufildr 21024	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 )
Theorem	fin1aufil 21025	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 = (/) ) )
12.2.4  Filter limits
Syntax	cfm 21026	Extend class definition to include the neighborhood filter mapping function.
class FilMap
Syntax	cflim 21027	Extend class notation with a function returning the limit of a filter.
class fLim
Syntax	cflf 21028	Extend class definition to include the function for filter-based function limits.
class fLimf
Syntax	cfcls 21029	Extend class definition to include the cluster point function on filters.
class fClus
Syntax	cfcf 21030	Extend class definition to include the function for cluster points of a function.
class fClusf
Definition	df-fm 21031*	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 ) ) ) ) )
Definition	df-flim 21032*	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 ) } )
Definition	df-flf 21033*	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 ) ) ) )
Definition	df-fcls 21034*	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 ) , (/) ) )
Definition	df-fcf 21035*	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 ) ) ) )
Theorem	fmval 21036*	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.)
|- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( X FilMap F ) ` B ) = ( X filGen ran ( y e. B |-> ( F " y ) ) ) )
Theorem	fmfil 21037	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 ) )
Theorem	fmf 21038	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 ) )
Theorem	fmss 21039	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 ) )
Theorem	elfm 21040*	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 ) ) )
Theorem	elfm2 21041*	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 ) ) )
Theorem	fmfg 21042	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 ) )
Theorem	elfm3 21043*	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 ) ) )
Theorem	imaelfm 21044	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 ) )
Theorem	rnelfmlem 21045*	Lemma for rnelfm 21046. (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 ) )
Theorem	rnelfm 21046	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 ) )
Theorem	fmfnfmlem1 21047*	Lemma for fmfnfm 21051. (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 ) ) ) )
Theorem	fmfnfmlem2 21048*	Lemma for fmfnfm 21051. (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 ) ) ) )
Theorem	fmfnfmlem3 21049*	Lemma for fmfnfm 21051. (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 ) ) )
Theorem	fmfnfmlem4 21050*	Lemma for fmfnfm 21051. (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 ) ) )
Theorem	fmfnfm 21051*	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 ) ) )
Theorem	fmufil 21052	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 ) )
Theorem	fmid 21053	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 )
Theorem	fmco 21054	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 ) ) )
Theorem	ufldom 21055	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 )
Theorem	flimval 21056*	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 ) } )
Theorem	elflim2 21057	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 ) ) )
Theorem	flimtop 21058	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 )
Theorem	flimneiss 21059	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 )
Theorem	flimnei 21060	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 )
Theorem	flimelbas 21061	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 )
Theorem	flimfil 21062	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 ) )
Theorem	flimtopon 21063	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 ) ) )
Theorem	elflim 21064	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 ) ) )
Theorem	flimss2 21065	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 ) )
Theorem	flimss1 21066	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 ) )
Theorem	neiflim 21067	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 } ) ) )
Theorem	flimopn 21068*	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 ) ) ) )
Theorem	fbflim 21069*	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 ) ) ) )
Theorem	fbflim2 21070*	A condition for a filter base B to converge to a point A. Use neighborhoods instead of open neighborhoods. Compare fbflim 21069. (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 ) ) )
Theorem	flimclsi 21071	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 ) )
Theorem	hausflimlem 21072	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 ) =/= (/) )
Theorem	hausflimi 21073*	One direction of hausflim 21074. A filter in a Hausdorff 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 ) )
Theorem	hausflim 21074*	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 ) ) )
Theorem	flimcf 21075*	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 ) ) )
Theorem	flimrest 21076	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 ) -> ( ( J ↾t Y ) fLim ( F ↾t Y ) ) = ( ( J fLim F ) i^i Y ) )
Theorem	flimclslem 21077	Lemma for flimcls 21078. (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 ) ) )
Theorem	flimcls 21078*	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 ) ) ) )
Theorem	flimsncls 21079	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 ) )
Theorem	hauspwpwf1 21080*	Lemma for hauspwpwdom 21081. 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 )
Theorem	hauspwpwdom 21081	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 )
Theorem	flffval 21082*	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 ) ) ) )
Theorem	flfval 21083	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.)
|- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( J fLimf L ) ` F ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
Theorem	flfnei 21084*	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.)
|- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A e. ( ( J fLimf L ) ` F ) <-> ( A e. X /\ A. n e. ( ( nei ` J ) ` { A } ) E. s e. L ( F " s ) C_ n ) ) )
Theorem	flfneii 21085*	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.)
|- X = U. J   =>    |- ( ( ( J e. Top /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ A e. ( ( J fLimf L ) ` F ) /\ N e. ( ( nei ` J ) ` { A } ) ) -> E. s e. L ( F " s ) C_ N )
Theorem	isflf 21086*	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.)
|- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A e. ( ( J fLimf L ) ` F ) <-> ( A e. X /\ A. o e. J ( A e. o -> E. s e. L ( F " s ) C_ o ) ) ) )
Theorem	flfelbas 21087	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.)
|- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ A e. ( ( J fLimf L ) ` F ) ) -> A e. X )
Theorem	flffbas 21088*	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.)
|- L = ( Y filGen B )   =>    |- ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( A e. ( ( J fLimf L ) ` F ) <-> ( A e. X /\ A. o e. J ( A e. o -> E. s e. B ( F " s ) C_ o ) ) ) )
Theorem	flftg 21089*	Limit points of a function can be defined using topological bases. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- J = ( topGen ` B )   =>    |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A e. ( ( J fLimf L ) ` F ) <-> ( A e. X /\ A. o e. B ( A e. o -> E. s e. L ( F " s ) C_ o ) ) ) )
Theorem	hausflf 21090*	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.)
|- X = U. J   =>    |- ( ( J e. Haus /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> E* x x e. ( ( J fLimf L ) ` F ) )
Theorem	hausflf2 21091	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.)
|- X = U. J   =>    |- ( ( ( J e. Haus /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ ( ( J fLimf L ) ` F ) =/= (/) ) -> ( ( J fLimf L ) ` F ) ~~ 1o )
Theorem	cnpflfi 21092	Forward direction of cnpflf 21094. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
|- ( ( A e. ( J fLim L ) /\ F e. ( ( J CnP K ) ` A ) ) -> ( F ` A ) e. ( ( K fLimf L ) ` F ) )
Theorem	cnpflf2 21093	F is continuous at point A iff a limit of F when x tends to A is ( F ` A ). Proposition 9 of [BourbakiTop1] p. TG I.50. (Contributed by FL, 29-May-2011.) (Revised by Mario Carneiro, 9-Apr-2015.)
|- L = ( ( nei ` J ) ` { A } )   =>    |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) -> ( F e. ( ( J CnP K ) ` A ) <-> ( F : X --> Y /\ ( F ` A ) e. ( ( K fLimf L ) ` F ) ) ) )
Theorem	cnpflf 21094*	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.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) -> ( F e. ( ( J CnP K ) ` A ) <-> ( F : X --> Y /\ A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) ) ) )
Theorem	cnflf 21095*	A function is continuous iff it respects filter limits. (Contributed by Jeff Hankins, 6-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. f e. ( Fil ` X ) A. x e. ( J fLim f ) ( F ` x ) e. ( ( K fLimf f ) ` F ) ) ) )
Theorem	cnflf2 21096*	A function is continuous iff it respects filter limits. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. f e. ( Fil ` X ) ( F " ( J fLim f ) ) C_ ( ( K fLimf f ) ` F ) ) ) )
Theorem	flfcnp 21097	A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ ( A e. ( ( J fLimf L ) ` F ) /\ G e. ( ( J CnP K ) ` A ) ) ) -> ( G ` A ) e. ( ( K fLimf L ) ` ( G o. F ) ) )
Theorem	lmflf 21098	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.)
|- Z = ( ZZ>= ` M )   &    |- L = ( Z filGen ( ZZ>= " Z ) )   =>    |- ( ( J e. (TopOn ` X ) /\ M e. ZZ /\ F : Z --> X ) -> ( F ( ~~> t ` J ) P <-> P e. ( ( J fLimf L ) ` F ) ) )
Theorem	txflf 21099*	Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> L e. ( Fil ` Z ) )   &    |- ( ph -> F : Z --> X )   &    |- ( ph -> G : Z --> Y )   &    |- H = ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. )   =>    |- ( ph -> ( <. R , S >. e. ( ( ( J tX K ) fLimf L ) ` H ) <-> ( R e. ( ( J fLimf L ) ` F ) /\ S e. ( ( K fLimf L ) ` G ) ) ) )
Theorem	flfcnp2 21100*	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.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> L e. ( Fil ` Z ) )   &    |- ( ( ph /\ x e. Z ) -> A e. X )   &    |- ( ( ph /\ x e. Z ) -> B e. Y )   &    |- ( ph -> R e. ( ( J fLimf L ) ` ( x e. Z |-> A ) ) )   &    |- ( ph -> S e. ( ( K fLimf L ) ` ( x e. Z |-> B ) ) )   &    |- ( ph -> O e. ( ( ( J tX K ) CnP N ) ` <. R , S >. ) )   =>    |- ( ph -> ( R O S ) e. ( ( N fLimf L ) ` ( x e. Z |-> ( A O B ) ) ) )