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	xpstopnlem1 20901*	The function F used in xpsval 15556 is a homeomorphism from the binary product topology to the indexed product topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- F = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )   &    |- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   =>    |- ( ph -> F e. ( ( J tX K ) Homeo ( Xt_ ` `' ( { J } +c { K } ) ) ) )
Theorem	xpstps 20902	A binary product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
|- T = ( R X.s S )   =>    |- ( ( R e. TopSp /\ S e. TopSp ) -> T e. TopSp )
Theorem	xpstopnlem2 20903*	Lemma for xpstopn 20904. (Contributed by Mario Carneiro, 27-Aug-2015.)
|- T = ( R X.s S )   &    |- J = ( TopOpen ` R )   &    |- K = ( TopOpen ` S )   &    |- O = ( TopOpen ` T )   &    |- X = ( Base ` R )   &    |- Y = ( Base ` S )   &    |- F = ( x e. X , y e. Y |-> `' ( { x } +c { y } ) )   =>    |- ( ( R e. TopSp /\ S e. TopSp ) -> O = ( J tX K ) )
Theorem	xpstopn 20904	The topology on a binary product of topological spaces, as we have defined it (transferring the indexed product topology on functions on { (/) , 1o } to ( X X. Y ) by the canonical bijection), coincides with the usual topological product (generated by a base of rectangles). (Contributed by Mario Carneiro, 27-Aug-2015.)
|- T = ( R X.s S )   &    |- J = ( TopOpen ` R )   &    |- K = ( TopOpen ` S )   &    |- O = ( TopOpen ` T )   =>    |- ( ( R e. TopSp /\ S e. TopSp ) -> O = ( J tX K ) )
Theorem	ptcmpfi 20905	A topological product of finitely many compact spaces is compact. This weak version of Tychonoff's theorem does not require the axiom of choice. (Contributed by Mario Carneiro, 8-Feb-2015.)
|- ( ( A e. Fin /\ F : A --> Comp ) -> ( Xt_ ` F ) e. Comp )
Theorem	xkocnv 20906*	The inverse of the "currying" function F is the uncurrying function. (Contributed by Mario Carneiro, 13-Apr-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- F = ( f e. ( ( J tX K ) Cn L ) |-> ( x e. X |-> ( y e. Y |-> ( x f y ) ) ) )   &    |- ( ph -> J e. 𝑛Locally Comp )   &    |- ( ph -> K e. 𝑛Locally Comp )   &    |- ( ph -> L e. Top )   =>    |- ( ph -> `' F = ( g e. ( J Cn ( L ^ko K ) ) |-> ( x e. X , y e. Y |-> ( ( g ` x ) ` y ) ) ) )
Theorem	xkohmeo 20907*	The Exponential Law for topological spaces. The "currying" function F is a homeomorphism on function spaces when J and K are exponentiable spaces (by xkococn 20752, it is sufficient to assume that J , K are locally compact to ensure exponentiability). (Contributed by Mario Carneiro, 13-Apr-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- F = ( f e. ( ( J tX K ) Cn L ) |-> ( x e. X |-> ( y e. Y |-> ( x f y ) ) ) )   &    |- ( ph -> J e. 𝑛Locally Comp )   &    |- ( ph -> K e. 𝑛Locally Comp )   &    |- ( ph -> L e. Top )   =>    |- ( ph -> F e. ( ( L ^ko ( J tX K ) ) Homeo ( ( L ^ko K ) ^ko J ) ) )
Theorem	qtopf1 20908	If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> F : X -1-1-> Y )   =>    |- ( ph -> F e. ( J Homeo ( J qTop F ) ) )
Theorem	qtophmeo 20909*	If two functions on a base topology J make the same identifications in order to create quotient spaces J qTop F and J qTop G, then not only are J qTop F and J qTop G homeomorphic, but there is a unique homeomorphism that makes the diagram commute. (Contributed by Mario Carneiro, 24-Mar-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> F : X -onto-> Y )   &    |- ( ph -> G : X -onto-> Y )   &    |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( ( F ` x ) = ( F ` y ) <-> ( G ` x ) = ( G ` y ) ) )   =>    |- ( ph -> E! f e. ( ( J qTop F ) Homeo ( J qTop G ) ) G = ( f o. F ) )
Theorem	t0kq 20910*	A topological space is T0 iff the quotient map is a homeomorphism onto the space's Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- F = ( x e. X |-> { y e. J | x e. y } )   =>    |- ( J e. (TopOn ` X ) -> ( J e. Kol2 <-> F e. ( J Homeo (KQ ` J ) ) ) )
Theorem	kqhmph 20911	A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J e. Kol2 <-> J ~= (KQ ` J ) )
Theorem	ist1-5lem 20912	Lemma for ist1-5 20914 and similar theorems. If A is a topological property which implies T0, such as T1 or T2, the property can be "decomposed" into T0 and a non-T0 version of property A (which is defined as stating that the Kolmogorov quotient of the space has property A). For example, if A is T1, then the theorem states that a space is T1 iff it is T0 and its Kolmogorov quotient is T1 (we call this property R0). (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J e. A -> J e. Kol2 )   &    |- ( J ~= (KQ ` J ) -> ( J e. A -> (KQ ` J ) e. A ) )   &    |- ( (KQ ` J ) ~= J -> ( (KQ ` J ) e. A -> J e. A ) )   =>    |- ( J e. A <-> ( J e. Kol2 /\ (KQ ` J ) e. A ) )
Theorem	t1r0 20913	A T1 space is R0. That is, the Kolmogorov quotient of a T1 space is also T1 (because they are homeomorphic). (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J e. Fre -> (KQ ` J ) e. Fre )
Theorem	ist1-5 20914	A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J e. Fre <-> ( J e. Kol2 /\ (KQ ` J ) e. Fre ) )
Theorem	ishaus3 20915	A topological space is Hausdorff iff it is both T0 and R1 (where R1 means that any two topologically distinct points are separated by neighborhoods). (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J e. Haus <-> ( J e. Kol2 /\ (KQ ` J ) e. Haus ) )
Theorem	nrmreg 20916	A normal T1 space is regular Hausdorff. In other words, a T4 space is T3 . One can get away with slightly weaker assumptions; see nrmr0reg 20841. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( ( J e. Nrm /\ J e. Fre ) -> J e. Reg )
Theorem	reghaus 20917	A regular T0 space is Hausdorff. In other words, a T3 space is T2 . A regular Hausdorff or T0 space is also known as a T3 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- ( J e. Reg -> ( J e. Haus <-> J e. Kol2 ) )
Theorem	nrmhaus 20918	A T1 normal space is Hausdorff. A Hausdorff or T1 normal space is also known as a T4 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- ( J e. Nrm -> ( J e. Haus <-> J e. Fre ) )
12.2  Filters and filter bases
12.2.1  Filter bases
Theorem	elmptrab 20919*	Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
|- F = ( x e. D |-> { y e. B | ph } )   &    |- ( ( x = X /\ y = Y ) -> ( ph <-> ps ) )   &    |- ( x = X -> B = C )   &    |- ( x e. D -> B e. V )   =>    |- ( Y e. ( F ` X ) <-> ( X e. D /\ Y e. C /\ ps ) )
Theorem	elmptrab2OLD 20920*	Obsolete version of elmptrab2 20921 as of 26-Mar-2021. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F = ( x e. _V |-> { y e. B | ph } )   &    |- ( ( x = X /\ y = Y ) -> ( ph <-> ps ) )   &    |- ( x = X -> B = C )   &    |- B e. V   &    |- ( Y e. C -> X e. W )   =>    |- ( Y e. ( F ` X ) <-> ( Y e. C /\ ps ) )
Theorem	elmptrab2 20921*	Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Revised by AV, 26-Mar-2021.)
|- F = ( x e. _V |-> { y e. B | ph } )   &    |- ( ( x = X /\ y = Y ) -> ( ph <-> ps ) )   &    |- ( x = X -> B = C )   &    |- B e. _V   &    |- ( Y e. C -> X e. W )   =>    |- ( Y e. ( F ` X ) <-> ( Y e. C /\ ps ) )
Theorem	isfbas 20922*	The predicate " F is a filter base." Note that some authors require filter bases to be closed under pairwise intersections, but that is not necessary under our definition. One advantage of this definition is that tails in a directed set form a filter base under our meaning. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
|- ( B e. A -> ( F e. ( fBas ` B ) <-> ( F C_ ~P B /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F ( F i^i ~P ( x i^i y ) ) =/= (/) ) ) ) )
Theorem	fbasne0 20923	There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
|- ( F e. ( fBas ` B ) -> F =/= (/) )
Theorem	0nelfb 20924	No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
|- ( F e. ( fBas ` B ) -> -. (/) e. F )
Theorem	fbsspw 20925	A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
|- ( F e. ( fBas ` B ) -> F C_ ~P B )
Theorem	fbelss 20926	An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
|- ( ( F e. ( fBas ` B ) /\ X e. F ) -> X C_ B )
Theorem	fbdmn0 20927	The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( F e. ( fBas ` B ) -> B =/= (/) )
Theorem	isfbas2 20928*	The predicate " F is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( B e. A -> ( F e. ( fBas ` B ) <-> ( F C_ ~P B /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F E. z e. F z C_ ( x i^i y ) ) ) ) )
Theorem	fbasssin 20929*	A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)
|- ( ( F e. ( fBas ` X ) /\ A e. F /\ B e. F ) -> E. x e. F x C_ ( A i^i B ) )
Theorem	fbssfi 20930*	A filter base contains subsets of its finite intersections. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( ( F e. ( fBas ` X ) /\ A e. ( fi ` F ) ) -> E. x e. F x C_ A )
Theorem	fbssint 20931*	A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( ( F e. ( fBas ` B ) /\ A C_ F /\ A e. Fin ) -> E. x e. F x C_ |^| A )
Theorem	fbncp 20932	A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( ( F e. ( fBas ` X ) /\ A e. F ) -> -. ( B \ A ) e. F )
Theorem	fbun 20933*	A necessary and sufficient condition for the union of two filter bases to also be a filter base. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) ) -> ( ( F u. G ) e. ( fBas ` X ) <-> A. x e. F A. y e. G E. z e. ( F u. G ) z C_ ( x i^i y ) ) )
Theorem	fbfinnfr 20934	No filter base containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( ( F e. ( fBas ` B ) /\ S e. F /\ S e. Fin ) -> |^| F =/= (/) )
Theorem	opnfbas 20935*	The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> { x e. J | S C_ x } e. ( fBas ` X ) )
Theorem	trfbas2 20936	Conditions for the trace of a filter base F to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
|- ( ( F e. ( fBas ` Y ) /\ A C_ Y ) -> ( ( F ↾t A ) e. ( fBas ` A ) <-> -. (/) e. ( F ↾t A ) ) )
Theorem	trfbas 20937*	Conditions for the trace of a filter base F to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
|- ( ( F e. ( fBas ` Y ) /\ A C_ Y ) -> ( ( F ↾t A ) e. ( fBas ` A ) <-> A. v e. F ( v i^i A ) =/= (/) ) )
12.2.2  Filters
Syntax	cfil 20938	Extend class notation with the set of filters on a set.
class Fil
Definition	df-fil 20939*	The set of filters on a set. Definition 1 (axioms FI, FIIa, FIIb, FIII) of [BourbakiTop1] p. I.36. Filters are used to define the concept of limit in the general case. They are a generalization of the idea of neighborhoods. Suppose you are in RR. With neighborhoods you can express the idea of a variable that tends to a specific number but you can't express the idea of a variable that tends to infinity. Filters relax the "axioms" of neighborhoods and then succeed in expressing the idea of something that tends to infinity. Filters were invented by Cartan in 1937 and made famous by Bourbaki in his treatise. A notion similar to the notion of filter is the concept of net invented by Moore and Smith in 1922. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- Fil = ( z e. _V |-> { f e. ( fBas ` z ) | A. x e. ~P z ( ( f i^i ~P x ) =/= (/) -> x e. f ) } )
Theorem	isfil 20940*	The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
|- ( F e. ( Fil ` X ) <-> ( F e. ( fBas ` X ) /\ A. x e. ~P X ( ( F i^i ~P x ) =/= (/) -> x e. F ) ) )
Theorem	filfbas 20941	A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
|- ( F e. ( Fil ` X ) -> F e. ( fBas ` X ) )
Theorem	0nelfil 20942	The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
|- ( F e. ( Fil ` X ) -> -. (/) e. F )
Theorem	fileln0 20943	An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)
|- ( ( F e. ( Fil ` X ) /\ A e. F ) -> A =/= (/) )
Theorem	filsspw 20944	A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
|- ( F e. ( Fil ` X ) -> F C_ ~P X )
Theorem	filelss 20945	An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
|- ( ( F e. ( Fil ` X ) /\ A e. F ) -> A C_ X )
Theorem	filss 20946	A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( ( F e. ( Fil ` X ) /\ ( A e. F /\ B C_ X /\ A C_ B ) ) -> B e. F )
Theorem	filin 20947	A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( ( F e. ( Fil ` X ) /\ A e. F /\ B e. F ) -> ( A i^i B ) e. F )
Theorem	filtop 20948	The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( F e. ( Fil ` X ) -> X e. F )
Theorem	isfil2 20949*	Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. ( Fil ` X ) <-> ( ( F C_ ~P X /\ -. (/) e. F /\ X e. F ) /\ A. x e. ~P X ( E. y e. F y C_ x -> x e. F ) /\ A. x e. F A. y e. F ( x i^i y ) e. F ) )
Theorem	isfildlem 20950*	Lemma for isfild 20951. (Contributed by Mario Carneiro, 1-Dec-2013.)
|- ( ph -> ( x e. F <-> ( x C_ A /\ ps ) ) )   &    |- ( ph -> A e. _V )   =>    |- ( ph -> ( B e. F <-> ( B C_ A /\ [. B / x ]. ps ) ) )
Theorem	isfild 20951*	Sufficient condition for a set of the form { x e. ~P A | ph } to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( ph -> ( x e. F <-> ( x C_ A /\ ps ) ) )   &    |- ( ph -> A e. _V )   &    |- ( ph -> [. A / x ]. ps )   &    |- ( ph -> -. [. (/) / x ]. ps )   &    |- ( ( ph /\ y C_ A /\ z C_ y ) -> ( [. z / x ]. ps -> [. y / x ]. ps ) )   &    |- ( ( ph /\ y C_ A /\ z C_ A ) -> ( ( [. y / x ]. ps /\ [. z / x ]. ps ) -> [. ( y i^i z ) / x ]. ps ) )   =>    |- ( ph -> F e. ( Fil ` A ) )
Theorem	filfi 20952	A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( F e. ( Fil ` X ) -> ( fi ` F ) = F )
Theorem	filinn0 20953	The intersection of two elements of a filter can't be empty. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( ( F e. ( Fil ` X ) /\ A e. F /\ B e. F ) -> ( A i^i B ) =/= (/) )
Theorem	filintn0 20954	A filter has the finite intersection property. Remark below definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 20-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( ( F e. ( Fil ` X ) /\ ( A C_ F /\ A =/= (/) /\ A e. Fin ) ) -> |^| A =/= (/) )
Theorem	filn0 20955	The empty set is not a filter. Remark below def. 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
|- ( F e. ( Fil ` X ) -> F =/= (/) )
Theorem	infil 20956	The intersection of two filters is a filter. Use fiint 7866 to extend this property to the intersection of a finite set of filters. Paragraph 3 of [BourbakiTop1] p. I.36. (Contributed by FL, 17-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( ( F e. ( Fil ` X ) /\ G e. ( Fil ` X ) ) -> ( F i^i G ) e. ( Fil ` X ) )
Theorem	snfil 20957	A singleton is a filter. Example 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( ( A e. B /\ A =/= (/) ) -> { A } e. ( Fil ` A ) )
Theorem	fbasweak 20958	A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)
|- ( ( F e. ( fBas ` X ) /\ F C_ ~P Y /\ Y e. V ) -> F e. ( fBas ` Y ) )
Theorem	snfbas 20959	Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
|- ( ( A C_ B /\ A =/= (/) /\ B e. V ) -> { A } e. ( fBas ` B ) )
Theorem	fsubbas 20960	A condition for a set to generate a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( X e. V -> ( ( fi ` A ) e. ( fBas ` X ) <-> ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) )
Theorem	fbasfip 20961	A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. ( fBas ` X ) -> -. (/) e. ( fi ` F ) )
Theorem	fbunfip 20962*	A helpful lemma for showing that certain sets generate filters. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` Y ) ) -> ( -. (/) e. ( fi ` ( F u. G ) ) <-> A. x e. F A. y e. G ( x i^i y ) =/= (/) ) )
Theorem	fgval 20963*	The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. ( fBas ` X ) -> ( X filGen F ) = { x e. ~P X | ( F i^i ~P x ) =/= (/) } )
Theorem	elfg 20964*	A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. ( fBas ` X ) -> ( A e. ( X filGen F ) <-> ( A C_ X /\ E. x e. F x C_ A ) ) )
Theorem	ssfg 20965	A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. ( fBas ` X ) -> F C_ ( X filGen F ) )
Theorem	fgss 20966	A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( X filGen F ) C_ ( X filGen G ) )
Theorem	fgss2 20967*	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.)
|- ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) ) -> ( ( X filGen F ) C_ ( X filGen G ) <-> A. x e. F E. y e. G y C_ x ) )
Theorem	fgfil 20968	A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. ( Fil ` X ) -> ( X filGen F ) = F )
Theorem	elfilss 20969*	An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)
|- ( ( F e. ( Fil ` X ) /\ A C_ X ) -> ( A e. F <-> E. t e. F t C_ A ) )
Theorem	filfinnfr 20970	No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( ( F e. ( Fil ` X ) /\ S e. F /\ S e. Fin ) -> |^| F =/= (/) )
Theorem	fgcl 20971	A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. ( fBas ` X ) -> ( X filGen F ) e. ( Fil ` X ) )
Theorem	fgabs 20972	Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)
|- ( ( F e. ( fBas ` Y ) /\ Y C_ X ) -> ( X filGen ( Y filGen F ) ) = ( X filGen F ) )
Theorem	neifil 20973	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.)
|- ( ( J e. (TopOn ` X ) /\ S C_ X /\ S =/= (/) ) -> ( ( nei ` J ) ` S ) e. ( Fil ` X ) )
Theorem	filunibas 20974	Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. ( Fil ` X ) -> U. F = X )
Theorem	filunirn 20975	Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. U. ran Fil <-> F e. ( Fil ` U. F ) )
Theorem	filcon 20976	A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. ( Fil ` X ) -> ( F u. { (/) } ) e. Con )
Theorem	fbasrn 20977*	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.)
|- C = ran ( x e. B |-> ( F " x ) )   =>    |- ( ( B e. ( fBas ` X ) /\ F : X --> Y /\ Y e. V ) -> C e. ( fBas ` Y ) )
Theorem	filuni 20978*	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.)
|- ( ( F C_ ( Fil ` X ) /\ F =/= (/) /\ A. f e. F A. g e. F ( f u. g ) e. F ) -> U. F e. ( Fil ` X ) )
Theorem	trfil1 20979	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 20980*	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 20981	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 20982	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 20983	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 20984	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 20983 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 20985	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 20980 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 20986*	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 20987*	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 20988*	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 20989	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 20990	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 20991	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 20992	Extend class notation with the ultrafilters-on-a-set function.
class UFil
Syntax	cufl 20993	Extend class notation with the ultrafilter lemma.
class UFL
Definition	df-ufil 20994*	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 20995*	Define the class of base sets for which the ultrafilter lemma filssufil 21005 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 20996*	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 20997	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 20998	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 20999	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 21000	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 )