P. 209 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cmphmph 20801	Compactness is a topological property-that is, for any two homeomorphic topologies, either both are compact or neither is. (Contributed by Jeff Hankins, 30-Jun-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( J ~= K -> ( J e. Comp -> K e. Comp ) )
Theorem	conhmph 20802	Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.)
|- ( J ~= K -> ( J e. Con -> K e. Con ) )
Theorem	t0hmph 20803	T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J ~= K -> ( J e. Kol2 -> K e. Kol2 ) )
Theorem	t1hmph 20804	T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J ~= K -> ( J e. Fre -> K e. Fre ) )
Theorem	haushmph 20805	Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J ~= K -> ( J e. Haus -> K e. Haus ) )
Theorem	reghmph 20806	Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J ~= K -> ( J e. Reg -> K e. Reg ) )
Theorem	nrmhmph 20807	Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J ~= K -> ( J e. Nrm -> K e. Nrm ) )
Theorem	hmph0 20808	A topology homeomorphic to the empty set is empty. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
|- ( J ~= { (/) } <-> J = { (/) } )
Theorem	hmphdis 20809	Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- X = U. J   =>    |- ( J ~= ~P A -> J = ~P X )
Theorem	hmphindis 20810	Homeomorphisms preserve topological indiscretion. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
|- X = U. J   =>    |- ( J ~= { (/) , A } -> J = { (/) , X } )
Theorem	indishmph 20811	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 20812	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 20813	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 20814	Lemma for ordthmeo 20815. (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 20815	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 20816*	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 20817*	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 20818*	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 20819*	The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.)
|- 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 20820*	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 20821*	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 20822*	The function F used in xpsval 15477 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 20823	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 20824*	Lemma for xpstopn 20825. (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 20825	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 20826	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 20827*	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 20828*	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 20673, 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 20829	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 20830*	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 20831*	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 20832	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 20833	Lemma for ist1-5 20835 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 20834	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 20835	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 20836	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 20837	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 20762. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( ( J e. Nrm /\ J e. Fre ) -> J e. Reg )
Theorem	reghaus 20838	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 20839	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 20840*	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	elmptrab2 20841*	Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
|- 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 20842*	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 20843	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 20844	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 20845	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 20846	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 20847	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 20848*	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 20849*	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 20850*	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 20851*	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 20852	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 20853*	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 20854	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 20855*	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 20856	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 20857*	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 20858	Extend class notation with the set of filters on a set.
class Fil
Definition	df-fil 20859*	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 20860*	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 20861	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 20862	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 20863	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 20864	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 20865	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 20866	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 20867	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 20868	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 20869*	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 20870*	Lemma for isfild 20871. (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 20871*	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 20872	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 20873	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 20874	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 20875	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 20876	The intersection of two filters is a filter. Use fiint 7857 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 20877	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 20878	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 20879	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 20880	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 20881	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 20882*	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 20883*	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 20884*	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 20885	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 20886	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 20887*	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 20888	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 20889*	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 20890	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 20891	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 20892	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 20893	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 20894	Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
|- ( F e. ( Fil ` X ) -> U. F = X )
Theorem	filunirn 20895	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 20896	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 20897*	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 20898*	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 20899	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 20900*	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 ) =/= (/) ) )