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	hmeoqtop 20801	A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( F e. ( J Homeo K ) -> K = ( J qTop F ) )
Theorem	hmph 20802	Express the predicate J is homeomorphic to K. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( J ~= K <-> ( J Homeo K ) =/= (/) )
Theorem	hmphi 20803	If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( F e. ( J Homeo K ) -> J ~= K )
Theorem	hmphtop 20804	Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)
|- ( J ~= K -> ( J e. Top /\ K e. Top ) )
Theorem	hmphtop1 20805	The relation "being homeomorphic to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( J ~= K -> J e. Top )
Theorem	hmphtop2 20806	The relation "being homeomorphic to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( J ~= K -> K e. Top )
Theorem	hmphref 20807	"Is homeomorphic to" is reflexive. (Contributed by FL, 25-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|- ( J e. Top -> J ~= J )
Theorem	hmphsym 20808	"Is homeomorphic to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.)
|- ( J ~= K -> K ~= J )
Theorem	hmphtr 20809	"Is homeomorphic to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( ( J ~= K /\ K ~= L ) -> J ~= L )
Theorem	hmpher 20810	"Is homeomorphic to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ~= Er Top
Theorem	hmphen 20811	Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
|- ( J ~= K -> J ~~ K )
Theorem	hmphsymb 20812	"Is homeomorphic to" is symmetric. (Contributed by FL, 22-Feb-2007.)
|- ( J ~= K <-> K ~= J )
Theorem	haushmphlem 20813*	Lemma for haushmph 20818 and similar theorems. If the topological property A is preserved under injective preimages, then property A is preserved under homeomorphisms. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J e. A -> J e. Top )   &    |- ( ( J e. A /\ f : U. K -1-1-> U. J /\ f e. ( K Cn J ) ) -> K e. A )   =>    |- ( J ~= K -> ( J e. A -> K e. A ) )
Theorem	cmphmph 20814	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 20815	Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.)
|- ( J ~= K -> ( J e. Con -> K e. Con ) )
Theorem	t0hmph 20816	T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J ~= K -> ( J e. Kol2 -> K e. Kol2 ) )
Theorem	t1hmph 20817	T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J ~= K -> ( J e. Fre -> K e. Fre ) )
Theorem	haushmph 20818	Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J ~= K -> ( J e. Haus -> K e. Haus ) )
Theorem	reghmph 20819	Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J ~= K -> ( J e. Reg -> K e. Reg ) )
Theorem	nrmhmph 20820	Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( J ~= K -> ( J e. Nrm -> K e. Nrm ) )
Theorem	hmph0 20821	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 20822	Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- X = U. J   =>    |- ( J ~= ~P A -> J = ~P X )
Theorem	hmphindis 20823	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 20824	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 20825	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 20826	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 20827	Lemma for ordthmeo 20828. (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 20828	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 20829*	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 20830*	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 20831*	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 20832*	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 20833*	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 20834*	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 20835*	The function F used in xpsval 15489 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 20836	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 20837*	Lemma for xpstopn 20838. (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 20838	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 20839	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 20840*	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 20841*	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 20686, 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 20842	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 20843*	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 20844*	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 20845	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 20846	Lemma for ist1-5 20848 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 20847	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 20848	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 20849	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 20850	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 20775. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- ( ( J e. Nrm /\ J e. Fre ) -> J e. Reg )
Theorem	reghaus 20851	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 20852	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 20853*	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 20854*	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 20855*	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 20856	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 20857	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 20858	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 20859	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 20860	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 20861*	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 20862*	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 20863*	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 20864*	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 20865	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 20866*	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 20867	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 20868*	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 20869	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 20870*	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 20871	Extend class notation with the set of filters on a set.
class Fil
Definition	df-fil 20872*	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 20873*	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 20874	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 20875	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 20876	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 20877	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 20878	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 20879	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 20880	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 20881	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 20882*	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 20883*	Lemma for isfild 20884. (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 20884*	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 20885	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 20886	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 20887	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 20888	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 20889	The intersection of two filters is a filter. Use fiint 7835 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 20890	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 20891	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 20892	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 20893	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 20894	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 20895*	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 20896*	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 20897*	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 20898	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 20899	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 20900*	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 ) )