P. 210 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20901-21000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	trfil1 20901	Conditions for the trace of a filter L to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( ( L e. ( Fil ` Y ) /\ A C_ Y ) -> A = U. ( L ↾t A ) )
Theorem	trfil2 20902*	Conditions for the trace of a filter L to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( ( L e. ( Fil ` Y ) /\ A C_ Y ) -> ( ( L ↾t A ) e. ( Fil ` A ) <-> A. v e. L ( v i^i A ) =/= (/) ) )
Theorem	trfil3 20903	Conditions for the trace of a filter L to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
|- ( ( L e. ( Fil ` Y ) /\ A C_ Y ) -> ( ( L ↾t A ) e. ( Fil ` A ) <-> -. ( Y \ A ) e. L ) )
Theorem	trfilss 20904	If A is a member of the filter, then the filter truncated to A is a subset of the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
|- ( ( F e. ( Fil ` X ) /\ A e. F ) -> ( F ↾t A ) C_ F )
Theorem	fgtr 20905	If A is a member of the filter, then truncating F to A and regenerating the behavior outside A using filGen recovers the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
|- ( ( F e. ( Fil ` X ) /\ A e. F ) -> ( X filGen ( F ↾t A ) ) = F )
Theorem	trfg 20906	The trace operation and the filGen operation are inverses to one another in some sense, with filGen growing the base set and ↾t shrinking it. See fgtr 20905 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)
|- ( ( F e. ( Fil ` A ) /\ A C_ X /\ X e. V ) -> ( ( X filGen F ) ↾t A ) = F )
Theorem	trnei 20907	The trace, over a set A, of the filter of the neighborhoods of a point P is a filter iff P belongs to the closure of A. (This is trfil2 20902 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> ( P e. ( ( cls ` J ) ` A ) <-> ( ( ( nei ` J ) ` { P } ) ↾t A ) e. ( Fil ` A ) ) )
Theorem	cfinfil 20908*	Relative complements of the finite parts of an infinite set is a filter. When A = NN 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.)
|- ( ( X e. V /\ A C_ X /\ -. A e. Fin ) -> { x e. ~P X | ( A \ x ) e. Fin } e. ( Fil ` X ) )
Theorem	csdfil 20909*	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.)
|- ( ( X e. dom card /\ om ~<_ X ) -> { x e. ~P X | ( X \ x ) ~< X } e. ( Fil ` X ) )
Theorem	supfil 20910*	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.)
|- ( ( A e. V /\ B C_ A /\ B =/= (/) ) -> { x e. ~P A | B C_ x } e. ( Fil ` A ) )
Theorem	zfbas 20911	The set of upper sets of integers is a filter base on ZZ, which corresponds to convergence of sequences on ZZ. (Contributed by Mario Carneiro, 13-Oct-2015.)
|- ran ZZ>= e. ( fBas ` ZZ )
Theorem	uzrest 20912	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.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> ( ran ZZ>= ↾t Z ) = ( ZZ>= " Z ) )
Theorem	uzfbas 20913	The set of upper sets of integers based at a point in a fixed upper integer set like NN is a filter base on NN, which corresponds to convergence of sequences on NN. (Contributed by Mario Carneiro, 13-Oct-2015.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> ( ZZ>= " Z ) e. ( fBas ` Z ) )
12.2.3  Ultrafilters
Syntax	cufil 20914	Extend class notation with the ultrafilters-on-a-set function.
class UFil
Syntax	cufl 20915	Extend class notation with the ultrafilter lemma.
class UFL
Definition	df-ufil 20916*	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.)
|- UFil = ( g e. _V |-> { f e. ( Fil ` g ) | A. x e. ~P g ( x e. f \/ ( g \ x ) e. f ) } )
Definition	df-ufl 20917*	Define the class of base sets for which the ultrafilter lemma filssufil 20927 holds. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- UFL = { x | A. f e. ( Fil ` x ) E. g e. ( UFil ` x ) f C_ g }
Theorem	isufil 20918*	The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
|- ( F e. ( UFil ` X ) <-> ( F e. ( Fil ` X ) /\ A. x e. ~P X ( x e. F \/ ( X \ x ) e. F ) ) )
Theorem	ufilfil 20919	An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
|- ( F e. ( UFil ` X ) -> F e. ( Fil ` X ) )
Theorem	ufilss 20920	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.)
|- ( ( F e. ( UFil ` X ) /\ S C_ X ) -> ( S e. F \/ ( X \ S ) e. F ) )
Theorem	ufilb 20921	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.)
|- ( ( F e. ( UFil ` X ) /\ S C_ X ) -> ( -. S e. F <-> ( X \ S ) e. F ) )
Theorem	ufilmax 20922	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.)
|- ( ( F e. ( UFil ` X ) /\ G e. ( Fil ` X ) /\ F C_ G ) -> F = G )
Theorem	isufil2 20923*	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 20924	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 20925	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 20926*	A filter is contained in some ultrafilter. This version of filssufil 20927 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 20927*	A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 8900.) (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 20928*	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 20929*	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 20930	Consequence of filssufilg 20926: 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 20931	A finite set satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( X e. Fin -> X e. UFL )
Theorem	acufl 20932	The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- (CHOICE -> UFL = _V )
Theorem	ssufl 20933	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 20934*	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 20935*	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 20936*	Lemma for fixufil 20937 and uffixfr 20938. (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 20937*	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 20938*	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 20939*	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 20940	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 20941	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 20942*	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 20943*	An ultrafilter is free iff it contains the Fréchet filter cfinfil 20908 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 20944*	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 20945*	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 20946	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 20947	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 20948	Extend class definition to include the neighborhood filter mapping function.
class FilMap
Syntax	cflim 20949	Extend class notation with a function returning the limit of a filter.
class fLim
Syntax	cflf 20950	Extend class definition to include the function for filter-based function limits.
class fLimf
Syntax	cfcls 20951	Extend class definition to include the cluster point function on filters.
class fClus
Syntax	cfcf 20952	Extend class definition to include the function for cluster points of a function.
class fClusf
Definition	df-fm 20953*	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 20954*	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 20955*	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 20956*	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 20957*	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 20958*	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 20959	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 20960	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 20961	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 20962*	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 20963*	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 20964	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 20965*	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 20966	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 20967*	Lemma for rnelfm 20968. (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 20968	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 20969*	Lemma for fmfnfm 20973. (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 20970*	Lemma for fmfnfm 20973. (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 20971*	Lemma for fmfnfm 20973. (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 20972*	Lemma for fmfnfm 20973. (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 20973*	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 20974	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 20975	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 20976	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 20977	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 20978*	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 20979	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 20980	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 20981	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 20982	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 20983	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 20984	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 20985	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 20986	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 20987	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 20988	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 20989	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 20990*	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 20991*	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 20992*	A condition for a filter base B to converge to a point A. Use neighborhoods instead of open neighborhoods. Compare fbflim 20991. (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 20993	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 20994	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 20995*	One direction of hausflim 20996. 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 20996*	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 20997*	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 20998	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 20999	Lemma for flimcls 21000. (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 21000*	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 ) ) ) )