P. 217 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21601-21700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	indishmph 21601	Equinumerous sets equipped with their indiscrete topologies are homeomorphic (which means in that particular case that a segment is homeomorphic to a circle contrary to what Wikipedia claims). (Contributed by FL, 17-Aug-2008.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
|- ( A ~~ B -> { (/) , A } ~= { (/) , B } )
Theorem	hmphen2 21602	Homeomorphisms preserve the cardinality of the underlying sets. (Contributed by FL, 17-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
|- X = U. J   &    |- Y = U. K   =>    |- ( J ~= K -> X ~~ Y )
Theorem	cmphaushmeo 21603	A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. (Contributed by Mario Carneiro, 17-Feb-2015.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( J e. Comp /\ K e. Haus /\ F e. ( J Cn K ) ) -> ( F e. ( J Homeo K ) <-> F : X -1-1-onto-> Y ) )
Theorem	ordthmeolem 21604	Lemma for ordthmeo 21605. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- X = dom R   &    |- Y = dom S   =>    |- ( ( R e. V /\ S e. W /\ F Isom R , S ( X , Y ) ) -> F e. ( (ordTop ` R ) Cn (ordTop ` S ) ) )
Theorem	ordthmeo 21605	An order isomorphism is a homeomorphism on the respective order topologies. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- X = dom R   &    |- Y = dom S   =>    |- ( ( R e. V /\ S e. W /\ F Isom R , S ( X , Y ) ) -> F e. ( (ordTop ` R ) Homeo (ordTop ` S ) ) )
Theorem	txhmeo 21606*	Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- X = U. J   &    |- Y = U. K   &    |- ( ph -> F e. ( J Homeo L ) )   &    |- ( ph -> G e. ( K Homeo M ) )   =>    |- ( ph -> ( x e. X , y e. Y |-> <. ( F ` x ) , ( G ` y ) >. ) e. ( ( J tX K ) Homeo ( L tX M ) ) )
Theorem	txswaphmeolem 21607*	Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- ( ( y e. Y , x e. X |-> <. x , y >. ) o. ( x e. X , y e. Y |-> <. y , x >. ) ) = ( _I |` ( X X. Y ) )
Theorem	txswaphmeo 21608*	There is a homeomorphism from X X. Y to Y X. X. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( x e. X , y e. Y |-> <. y , x >. ) e. ( ( J tX K ) Homeo ( K tX J ) ) )
Theorem	pt1hmeo 21609*	The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.) (Proof shortened by AV, 18-Apr-2021.)
|- K = ( Xt_ ` { <. A , J >. } )   &    |- ( ph -> A e. V )   &    |- ( ph -> J e. (TopOn ` X ) )   =>    |- ( ph -> ( x e. X |-> { <. A , x >. } ) e. ( J Homeo K ) )
Theorem	ptuncnv 21610*	Exhibit the converse function of the map G which joins two product topologies on disjoint index sets. (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|- X = U. K   &    |- Y = U. L   &    |- J = ( Xt_ ` F )   &    |- K = ( Xt_ ` ( F |` A ) )   &    |- L = ( Xt_ ` ( F |` B ) )   &    |- G = ( x e. X , y e. Y |-> ( x u. y ) )   &    |- ( ph -> C e. V )   &    |- ( ph -> F : C --> Top )   &    |- ( ph -> C = ( A u. B ) )   &    |- ( ph -> ( A i^i B ) = (/) )   =>    |- ( ph -> `' G = ( z e. U. J |-> <. ( z |` A ) , ( z |` B ) >. ) )
Theorem	ptunhmeo 21611*	Define a homeomorphism from a binary product of indexed product topologies to an indexed product topology on the union of the index sets. This is the topological analogue of ( A ^ B ) x. ( A ^ C ) = A ^ ( B + C ). (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|- X = U. K   &    |- Y = U. L   &    |- J = ( Xt_ ` F )   &    |- K = ( Xt_ ` ( F |` A ) )   &    |- L = ( Xt_ ` ( F |` B ) )   &    |- G = ( x e. X , y e. Y |-> ( x u. y ) )   &    |- ( ph -> C e. V )   &    |- ( ph -> F : C --> Top )   &    |- ( ph -> C = ( A u. B ) )   &    |- ( ph -> ( A i^i B ) = (/) )   =>    |- ( ph -> G e. ( ( K tX L ) Homeo J ) )
Theorem	xpstopnlem1 21612*	The function F used in xpsval 16232 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 21613	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 21614*	Lemma for xpstopn 21615. (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 21615	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 21616	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 21617*	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 21618*	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 21463, 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 21619	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 21620*	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 21621*	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 21622	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 21623	Lemma for ist1-5 21625 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 21624	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 21625	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 21626	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 21627	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 21552. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( ( J e. Nrm /\ J e. Fre ) -> J e. Reg )
Theorem	reghaus 21628	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 21629	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 21630*	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 21631*	Obsolete version of elmptrab2 21632 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 21632*	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 21633*	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 21634	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 21635	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 21636	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 21637	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 21638	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 21639*	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 21640*	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 21641*	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 21642*	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 21643	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 21644*	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 21645	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 21646*	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 21647	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 21648*	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 21649	Extend class notation with the set of filters on a set.
class Fil
Definition	df-fil 21650*	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 21651*	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 21652	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 21653	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 21654	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 21655	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 21656	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 21657	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 21658	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 21659	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 21660*	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 21661*	Lemma for isfild 21662. (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 21662*	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 21663	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 21664	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 21665	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 21666	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 21667	The intersection of two filters is a filter. Use fiint 8237 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 21668	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 21669	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 21670	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 21671	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 21672	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 21673*	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 21674*	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 21675*	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 21676	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 21677	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 21678*	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 21679	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 21680*	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 21681	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 21682	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 21683	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 21684	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 21685	Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. ( Fil ` X ) -> U. F = X )
Theorem	filunirn 21686	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	filconn 21687	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. Conn )
Theorem	fbasrn 21688*	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 21689*	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 21690	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 21691*	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 21692	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 21693	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 21694	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 21695	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 21694 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 21696	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 21691 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 21697*	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 21698*	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 21699*	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 21700	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 )