P. 151 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15001-15100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	psrneg 15001*	The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- N = ( invg ` R )   &    |- B = ( Base ` S )   &    |- M = ( invg ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( M ` X ) = ( N o. X ) )
Theorem	psr1clfi 15002*	The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Ring )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- .0. = ( 0g ` R )   &    |- .1. = ( 1r ` R )   &    |- U = ( x e. D |-> if ( x = ( I X. { 0 } ) , .1. , .0. ) )   &    |- B = ( Base ` S )   =>    |- ( ph -> U e. B )
Theorem	reldmmpl 15003	The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- Rel dom mPoly
Theorem	mplvalcoe 15004*	Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) }   =>    |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )
Theorem	mplbascoe 15005*	Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) } )
Theorem	mplelbascoe 15006*	Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> ( X e. U <-> ( X e. B /\ E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = .0. ) ) ) )
Theorem	fnmpl 15007	mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
|- mPoly Fn ( _V X. _V )
Theorem	mplrcl 15008	Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   =>    |- ( X e. B -> I e. _V )
Theorem	mplval2g 15009	Self-referential expression for the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )
Theorem	mplbasss 15010	The set of polynomials is a subset of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- U = ( Base ` P )   &    |- B = ( Base ` S )   =>    |- U C_ B
Theorem	mplelf 15011*	A polynomial is defined as a function on the coefficients. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- K = ( Base ` R )   &    |- B = ( Base ` P )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- ( ph -> X e. B )   =>    |- ( ph -> X : D --> K )
Theorem	mplsubgfilemm 15012*	Lemma for mplsubgfi 15015. There exists a polynomial. (Contributed by Jim Kingdon, 21-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> E. j j e. U )
Theorem	mplsubgfilemcl 15013	Lemma for mplsubgfi 15015. The sum of two polynomials is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- .+ = ( +g ` S )   =>    |- ( ph -> ( X .+ Y ) e. U )
Theorem	mplsubgfileminv 15014	Lemma for mplsubgfi 15015. The additive inverse of a polynomial is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- N = ( invg ` S )   =>    |- ( ph -> ( N ` X ) e. U )
Theorem	mplsubgfi 15015	The set of polynomials is closed under addition, i.e. it is a subgroup of the set of power series. (Contributed by Mario Carneiro, 8-Jan-2015.) (Proof shortened by AV, 16-Jul-2019.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> U e. (SubGrp ` S ) )
Theorem	mpl0fi 15016*	The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
|- P = ( I mPoly R )   &    |- O = ( 0g ` R )   &    |- .0. = ( 0g ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> .0. = ( x e. ( NN0 ^m I ) |-> O ) )
Theorem	mplplusgg 15017	Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- Y = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- .+ = ( +g ` Y )   =>    |- ( ( I e. V /\ R e. W ) -> .+ = ( +g ` S ) )
Theorem	mpladd 15018	The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   &    |- .+ = ( +g ` R )   &    |- .+b = ( +g ` P )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+b Y ) = ( X oF .+ Y ) )
Theorem	mplnegfi 15019	The negative function on multivariate polynomials. (Contributed by SN, 25-May-2024.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   &    |- N = ( invg ` R )   &    |- M = ( invg ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( M ` X ) = ( N o. X ) )
Theorem	mplgrpfi 15020	The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
|- P = ( I mPoly R )   =>    |- ( ( I e. Fin /\ R e. Grp ) -> P e. Grp )
PART 9  BASIC TOPOLOGY
9.1  Topology
9.1.1  Topological spaces A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set.
9.1.1.1  Topologies
Syntax	ctop 15021	Syntax for the class of topologies.
class Top
Definition	df-top 15022*	Define the class of topologies. It is a proper class. See istopg 15023 and istopfin 15024 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections. The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241. (Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)
|- Top = { x | ( A. y e. ~P x U. y e. x /\ A. y e. x A. z e. x ( y i^i z ) e. x ) }
Theorem	istopg 15023*	Express the predicate " J is a topology". See istopfin 15024 for another characterization using nonempty finite intersections instead of binary intersections. Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use T to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- ( J e. A -> ( J e. Top <-> ( A. x ( x C_ J -> U. x e. J ) /\ A. x e. J A. y e. J ( x i^i y ) e. J ) ) )
Theorem	istopfin 15024*	Express the predicate " J is a topology" using nonempty finite intersections instead of binary intersections as in istopg 15023. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
|- ( J e. Top -> ( A. x ( x C_ J -> U. x e. J ) /\ A. x ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> |^| x e. J ) ) )
Theorem	uniopn 15025	The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
|- ( ( J e. Top /\ A C_ J ) -> U. A e. J )
Theorem	iunopn 15026*	The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
|- ( ( J e. Top /\ A. x e. A B e. J ) -> U_ x e. A B e. J )
Theorem	inopn 15027	The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
|- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A i^i B ) e. J )
Theorem	fiinopn 15028	The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
|- ( J e. Top -> ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> |^| A e. J ) )
Theorem	unopn 15029	The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A u. B ) e. J )
Theorem	0opn 15030	The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
|- ( J e. Top -> (/) e. J )
Theorem	0ntop 15031	The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
|- -. (/) e. Top
Theorem	topopn 15032	The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
|- X = U. J   =>    |- ( J e. Top -> X e. J )
Theorem	eltopss 15033	A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
|- X = U. J   =>    |- ( ( J e. Top /\ A e. J ) -> A C_ X )
9.1.1.2  Topologies on sets
Syntax	ctopon 15034	Syntax for the function of topologies on sets.
class TopOn
Definition	df-topon 15035*	Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- TopOn = ( b e. _V |-> { j e. Top | b = U. j } )
Theorem	funtopon 15036	The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
|- Fun TopOn
Theorem	istopon 15037	Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) )
Theorem	topontop 15038	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) -> J e. Top )
Theorem	toponuni 15039	The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) -> B = U. J )
Theorem	topontopi 15040	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- J e. (TopOn ` B )   =>    |- J e. Top
Theorem	toponunii 15041	The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- J e. (TopOn ` B )   =>    |- B = U. J
Theorem	toptopon 15042	Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- X = U. J   =>    |- ( J e. Top <-> J e. (TopOn ` X ) )
Theorem	toptopon2 15043	A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
|- ( J e. Top <-> J e. (TopOn ` U. J ) )
Theorem	topontopon 15044	A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
|- ( J e. (TopOn ` X ) -> J e. (TopOn ` U. J ) )
Theorem	toponrestid 15045	Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
|- A e. (TopOn ` B )   =>    |- A = ( A ↾t B )
Theorem	toponsspwpwg 15046	The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
|- ( A e. V -> (TopOn ` A ) C_ ~P ~P A )
Theorem	dmtopon 15047	The domain of TopOn is _V. (Contributed by BJ, 29-Apr-2021.)
|- dom TopOn = _V
Theorem	fntopon 15048	The class TopOn is a function with domain _V. (Contributed by BJ, 29-Apr-2021.)
|- TopOn Fn _V
Theorem	toponmax 15049	The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) -> B e. J )
Theorem	toponss 15050	A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ A e. J ) -> A C_ X )
Theorem	toponcom 15051	If K is a topology on the base set of topology J, then J is a topology on the base of K. (Contributed by Mario Carneiro, 22-Aug-2015.)
|- ( ( J e. Top /\ K e. (TopOn ` U. J ) ) -> J e. (TopOn ` U. K ) )
Theorem	toponcomb 15052	Biconditional form of toponcom 15051. (Contributed by BJ, 5-Dec-2021.)
|- ( ( J e. Top /\ K e. Top ) -> ( J e. (TopOn ` U. K ) <-> K e. (TopOn ` U. J ) ) )
Theorem	topgele 15053	The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
|- ( J e. (TopOn ` X ) -> ( { (/) , X } C_ J /\ J C_ ~P X ) )
9.1.1.3  Topological spaces
Syntax	ctps 15054	Syntax for the class of topological spaces.
class TopSp
Definition	df-topsp 15055	Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
|- TopSp = { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) }
Theorem	istps 15056	Express the predicate "is a topological space". (Contributed by Mario Carneiro, 13-Aug-2015.)
|- A = ( Base ` K )   &    |- J = ( TopOpen ` K )   =>    |- ( K e. TopSp <-> J e. (TopOn ` A ) )
Theorem	istps2 15057	Express the predicate "is a topological space". (Contributed by NM, 20-Oct-2012.)
|- A = ( Base ` K )   &    |- J = ( TopOpen ` K )   =>    |- ( K e. TopSp <-> ( J e. Top /\ A = U. J ) )
Theorem	tpsuni 15058	The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
|- A = ( Base ` K )   &    |- J = ( TopOpen ` K )   =>    |- ( K e. TopSp -> A = U. J )
Theorem	tpstop 15059	The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)
|- J = ( TopOpen ` K )   =>    |- ( K e. TopSp -> J e. Top )
Theorem	tpspropd 15060	A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- ( ph -> ( Base ` K ) = ( Base ` L ) )   &    |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )   =>    |- ( ph -> ( K e. TopSp <-> L e. TopSp ) )
Theorem	topontopn 15061	Express the predicate "is a topological space". (Contributed by Mario Carneiro, 13-Aug-2015.)
|- A = ( Base ` K )   &    |- J = (TopSet ` K )   =>    |- ( J e. (TopOn ` A ) -> J = ( TopOpen ` K ) )
Theorem	tsettps 15062	If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
|- A = ( Base ` K )   &    |- J = (TopSet ` K )   =>    |- ( J e. (TopOn ` A ) -> K e. TopSp )
Theorem	istpsi 15063	Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
|- ( Base ` K ) = A   &    |- ( TopOpen ` K ) = J   &    |- A = U. J   &    |- J e. Top   =>    |- K e. TopSp
Theorem	eltpsg 15064	Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
|- K = { <. ( Base ` ndx ) , A >. , <. (TopSet ` ndx ) , J >. }   =>    |- ( J e. (TopOn ` A ) -> K e. TopSp )
Theorem	eltpsi 15065	Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- K = { <. ( Base ` ndx ) , A >. , <. (TopSet ` ndx ) , J >. }   &    |- A = U. J   &    |- J e. Top   =>    |- K e. TopSp
9.1.2  Topological bases
Syntax	ctb 15066	Syntax for the class of topological bases.
class TopBases
Definition	df-bases 15067*	Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 15069). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.)
|- TopBases = { x | A. y e. x A. z e. x ( y i^i z ) C_ U. ( x i^i ~P ( y i^i z ) ) }
Theorem	isbasisg 15068*	Express the predicate "the set B is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
|- ( B e. C -> ( B e. TopBases <-> A. x e. B A. y e. B ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) ) )
Theorem	isbasis2g 15069*	Express the predicate "the set B is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
|- ( B e. C -> ( B e. TopBases <-> A. x e. B A. y e. B A. z e. ( x i^i y ) E. w e. B ( z e. w /\ w C_ ( x i^i y ) ) ) )
Theorem	isbasis3g 15070*	Express the predicate "the set B is a basis for a topology". Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)
|- ( B e. C -> ( B e. TopBases <-> ( A. x e. B x C_ U. B /\ A. x e. U. B E. y e. B x e. y /\ A. x e. B A. y e. B A. z e. ( x i^i y ) E. w e. B ( z e. w /\ w C_ ( x i^i y ) ) ) ) )
Theorem	basis1 15071	Property of a basis. (Contributed by NM, 16-Jul-2006.)
|- ( ( B e. TopBases /\ C e. B /\ D e. B ) -> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) )
Theorem	basis2 15072*	Property of a basis. (Contributed by NM, 17-Jul-2006.)
|- ( ( ( B e. TopBases /\ C e. B ) /\ ( D e. B /\ A e. ( C i^i D ) ) ) -> E. x e. B ( A e. x /\ x C_ ( C i^i D ) ) )
Theorem	fiinbas 15073*	If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( B e. C /\ A. x e. B A. y e. B ( x i^i y ) e. B ) -> B e. TopBases )
Theorem	baspartn 15074*	A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( ( P e. V /\ A. x e. P A. y e. P ( x = y \/ ( x i^i y ) = (/) ) ) -> P e. TopBases )
Theorem	tgval2 15075*	Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 15088) that ( topGen ` B ) is indeed a topology (on U. B, see unitg 15086). See also tgval 13593 and tgval3 15082. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( topGen ` B ) = { x | ( x C_ U. B /\ A. y e. x E. z e. B ( y e. z /\ z C_ x ) ) } )
Theorem	eltg 15076	Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> A C_ U. ( B i^i ~P A ) ) )
Theorem	eltg2 15077*	Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> ( A C_ U. B /\ A. x e. A E. y e. B ( x e. y /\ y C_ A ) ) ) )
Theorem	eltg2b 15078*	Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> A. x e. A E. y e. B ( x e. y /\ y C_ A ) ) )
Theorem	eltg4i 15079	An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( A e. ( topGen ` B ) -> A = U. ( B i^i ~P A ) )
Theorem	eltg3i 15080	The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- ( ( B e. V /\ A C_ B ) -> U. A e. ( topGen ` B ) )
Theorem	eltg3 15081*	Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> E. x ( x C_ B /\ A = U. x ) ) )
Theorem	tgval3 15082*	Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 13593 and tgval2 15075. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
|- ( B e. V -> ( topGen ` B ) = { x | E. y ( y C_ B /\ x = U. y ) } )
Theorem	tg1 15083	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
|- ( A e. ( topGen ` B ) -> A C_ U. B )
Theorem	tg2 15084*	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
|- ( ( A e. ( topGen ` B ) /\ C e. A ) -> E. x e. B ( C e. x /\ x C_ A ) )
Theorem	bastg 15085	A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> B C_ ( topGen ` B ) )
Theorem	unitg 15086	The topology generated by a basis B is a topology on U. B. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)
|- ( B e. V -> U. ( topGen ` B ) = U. B )
Theorem	tgss 15087	Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
|- ( ( C e. V /\ B C_ C ) -> ( topGen ` B ) C_ ( topGen ` C ) )
Theorem	tgcl 15088	Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
|- ( B e. TopBases -> ( topGen ` B ) e. Top )
Theorem	tgclb 15089	The property tgcl 15088 can be reversed: if the topology generated by B is actually a topology, then B must be a topological basis. This yields an alternative definition of TopBases. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- ( B e. TopBases <-> ( topGen ` B ) e. Top )
Theorem	tgtopon 15090	A basis generates a topology on U. B. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( B e. TopBases -> ( topGen ` B ) e. (TopOn ` U. B ) )
Theorem	topbas 15091	A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
|- ( J e. Top -> J e. TopBases )
Theorem	tgtop 15092	A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
|- ( J e. Top -> ( topGen ` J ) = J )
Theorem	eltop 15093	Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
|- ( J e. Top -> ( A e. J <-> A C_ U. ( J i^i ~P A ) ) )
Theorem	eltop2 15094*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
|- ( J e. Top -> ( A e. J <-> A. x e. A E. y e. J ( x e. y /\ y C_ A ) ) )
Theorem	eltop3 15095*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
|- ( J e. Top -> ( A e. J <-> E. x ( x C_ J /\ A = U. x ) ) )
Theorem	tgdom 15096	A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)
|- ( B e. V -> ( topGen ` B ) ~<_ ~P B )
Theorem	tgiun 15097*	The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- ( ( B e. V /\ A. x e. A C e. B ) -> U_ x e. A C e. ( topGen ` B ) )
Theorem	tgidm 15098	The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
|- ( B e. V -> ( topGen ` ( topGen ` B ) ) = ( topGen ` B ) )
Theorem	bastop 15099	Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
|- ( B e. TopBases -> ( B e. Top <-> ( topGen ` B ) = B ) )
Theorem	tgtop11 15100	The topology generation function is one-to-one when applied to completed topologies. (Contributed by NM, 18-Jul-2006.)
|- ( ( J e. Top /\ K e. Top /\ ( topGen ` J ) = ( topGen ` K ) ) -> J = K )