P. 130 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12901-13000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	neiss2 12901	A set with a neighborhood is a subset of the base set of a topology. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.)
|- X = U. J   =>    |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )
Theorem	neival 12902*	Value of the set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X ) -> ( ( nei ` J ) ` S ) = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )
Theorem	isnei 12903*	The predicate "the class N is a neighborhood of S". (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
Theorem	neiint 12904	An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X /\ N C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> S C_ ( ( int ` J ) ` N ) ) )
Theorem	isneip 12905*	The predicate "the class N is a neighborhood of point P". (Contributed by NM, 26-Feb-2007.)
|- X = U. J   =>    |- ( ( J e. Top /\ P e. X ) -> ( N e. ( ( nei ` J ) ` { P } ) <-> ( N C_ X /\ E. g e. J ( P e. g /\ g C_ N ) ) ) )
Theorem	neii1 12906	A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.)
|- X = U. J   =>    |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> N C_ X )
Theorem	neisspw 12907	The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)
|- X = U. J   =>    |- ( J e. Top -> ( ( nei ` J ) ` S ) C_ ~P X )
Theorem	neii2 12908*	Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
Theorem	neiss 12909	Any neighborhood of a set S is also a neighborhood of any subset R C_ S. Similar to Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> N e. ( ( nei ` J ) ` R ) )
Theorem	ssnei 12910	A set is included in any of its neighborhoods. Generalization to subsets of elnei 12911. (Contributed by FL, 16-Nov-2006.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ N )
Theorem	elnei 12911	A point belongs to any of its neighborhoods. Property Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.)
|- ( ( J e. Top /\ P e. A /\ N e. ( ( nei ` J ) ` { P } ) ) -> P e. N )
Theorem	0nnei 12912	The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)
|- ( ( J e. Top /\ S =/= (/) ) -> -. (/) e. ( ( nei ` J ) ` S ) )
Theorem	neipsm 12913*	A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X /\ E. x x e. S ) -> ( N e. ( ( nei ` J ) ` S ) <-> A. p e. S N e. ( ( nei ` J ) ` { p } ) ) )
Theorem	opnneissb 12914	An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)
|- X = U. J   =>    |- ( ( J e. Top /\ N e. J /\ S C_ X ) -> ( S C_ N <-> N e. ( ( nei ` J ) ` S ) ) )
Theorem	opnssneib 12915	Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)
|- X = U. J   =>    |- ( ( J e. Top /\ S e. J /\ N C_ X ) -> ( S C_ N <-> N e. ( ( nei ` J ) ` S ) ) )
Theorem	ssnei2 12916	Any subset M of X containing a neighborhood N of a set S is a neighborhood of this set. Generalization to subsets of Property Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
|- X = U. J   =>    |- ( ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) /\ ( N C_ M /\ M C_ X ) ) -> M e. ( ( nei ` J ) ` S ) )
Theorem	opnneiss 12917	An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.)
|- ( ( J e. Top /\ N e. J /\ S C_ N ) -> N e. ( ( nei ` J ) ` S ) )
Theorem	opnneip 12918	An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.)
|- ( ( J e. Top /\ N e. J /\ P e. N ) -> N e. ( ( nei ` J ) ` { P } ) )
Theorem	tpnei 12919	The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 12917. (Contributed by FL, 2-Oct-2006.)
|- X = U. J   =>    |- ( J e. Top -> ( S C_ X <-> X e. ( ( nei ` J ) ` S ) ) )
Theorem	neiuni 12920	The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X ) -> X = U. ( ( nei ` J ) ` S ) )
Theorem	topssnei 12921	A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ J C_ K ) -> ( ( nei ` J ) ` S ) C_ ( ( nei ` K ) ` S ) )
Theorem	innei 12922	The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property Vii of [BourbakiTop1] p. I.3 for binary intersections. (Contributed by FL, 28-Sep-2006.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ M e. ( ( nei ` J ) ` S ) ) -> ( N i^i M ) e. ( ( nei ` J ) ` S ) )
Theorem	opnneiid 12923	Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)
|- ( J e. Top -> ( N e. ( ( nei ` J ) ` N ) <-> N e. J ) )
Theorem	neissex 12924*	For any neighborhood N of S, there is a neighborhood x of S such that N is a neighborhood of all subsets of x. Generalization to subsets of Property Viv of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. x e. ( ( nei ` J ) ` S ) A. y ( y C_ x -> N e. ( ( nei ` J ) ` y ) ) )
Theorem	0nei 12925	The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
|- ( J e. Top -> (/) e. ( ( nei ` J ) ` (/) ) )
8.1.6  Subspace topologies
Theorem	restrcl 12926	Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon, 23-Mar-2023.)
|- ( ( J ↾t A ) e. Top -> ( J e. _V /\ A e. _V ) )
Theorem	restbasg 12927	A subspace topology basis is a basis. (Contributed by Mario Carneiro, 19-Mar-2015.)
|- ( ( B e. TopBases /\ A e. V ) -> ( B ↾t A ) e. TopBases )
Theorem	tgrest 12928	A subspace can be generated by restricted sets from a basis for the original topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
|- ( ( B e. V /\ A e. W ) -> ( topGen ` ( B ↾t A ) ) = ( ( topGen ` B ) ↾t A ) )
Theorem	resttop 12929	A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89. A is normally a subset of the base set of J. (Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( ( J e. Top /\ A e. V ) -> ( J ↾t A ) e. Top )
Theorem	resttopon 12930	A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )
Theorem	restuni 12931	The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- X = U. J   =>    |- ( ( J e. Top /\ A C_ X ) -> A = U. ( J ↾t A ) )
Theorem	stoig 12932	The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
|- X = U. J   =>    |- ( ( J e. Top /\ A C_ X ) -> { <. ( Base ` ndx ) , A >. , <. (TopSet ` ndx ) , ( J ↾t A ) >. } e. TopSp )
Theorem	restco 12933	Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( ( J e. V /\ A e. W /\ B e. X ) -> ( ( J ↾t A ) ↾t B ) = ( J ↾t ( A i^i B ) ) )
Theorem	restabs 12934	Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
|- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( ( J ↾t T ) ↾t S ) = ( J ↾t S ) )
Theorem	restin 12935	When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)
|- X = U. J   =>    |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ( J ↾t ( A i^i X ) ) )
Theorem	restuni2 12936	The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- X = U. J   =>    |- ( ( J e. Top /\ A e. V ) -> ( A i^i X ) = U. ( J ↾t A ) )
Theorem	resttopon2 12937	The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( J ↾t A ) e. (TopOn ` ( A i^i X ) ) )
Theorem	rest0 12938	The subspace topology induced by the topology J on the empty set. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( J e. Top -> ( J ↾t (/) ) = { (/) } )
Theorem	restsn 12939	The only subspace topology induced by the topology { (/) }. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
|- ( A e. V -> ( { (/) } ↾t A ) = { (/) } )
Theorem	restopnb 12940	If B is an open subset of the subspace base set A, then any subset of B is open iff it is open in A. (Contributed by Mario Carneiro, 2-Mar-2015.)
|- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. J <-> C e. ( J ↾t A ) ) )
Theorem	ssrest 12941	If K is a finer topology than J, then the subspace topologies induced by A maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( ( K e. V /\ J C_ K ) -> ( J ↾t A ) C_ ( K ↾t A ) )
Theorem	restopn2 12942	If A is open, then B is open in A iff it is an open subset of A. (Contributed by Mario Carneiro, 2-Mar-2015.)
|- ( ( J e. Top /\ A e. J ) -> ( B e. ( J ↾t A ) <-> ( B e. J /\ B C_ A ) ) )
Theorem	restdis 12943	A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
|- ( ( A e. V /\ B C_ A ) -> ( ~P A ↾t B ) = ~P B )
8.1.7  Limits and continuity in topological spaces
Syntax	ccn 12944	Extend class notation with the class of continuous functions between topologies.
class Cn
Syntax	ccnp 12945	Extend class notation with the class of functions between topologies continuous at a given point.
class CnP
Syntax	clm 12946	Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.
class ~~> t
Definition	df-cn 12947*	Define a function on two topologies whose value is the set of continuous mappings from the first topology to the second. Based on definition of continuous function in [Munkres] p. 102. See iscn 12956 for the predicate form. (Contributed by NM, 17-Oct-2006.)
|- Cn = ( j e. Top , k e. Top |-> { f e. ( U. k ^m U. j ) | A. y e. k ( `' f " y ) e. j } )
Definition	df-cnp 12948*	Define a function on two topologies whose value is the set of continuous mappings at a specified point in the first topology. Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.)
|- CnP = ( j e. Top , k e. Top |-> ( x e. U. j |-> { f e. ( U. k ^m U. j ) | A. y e. k ( ( f ` x ) e. y -> E. g e. j ( x e. g /\ ( f " g ) C_ y ) ) } ) )
Definition	df-lm 12949*	Define a function on topologies whose value is the convergence relation for sequences into the given topological space. Although f is typically a sequence (a function from an upperset of integers) with values in the topological space, it need not be. Note, however, that the limit property concerns only values at integers, so that the real-valued function ( x e. RR |-> ( sin ` ( pi x. x ) ) ) converges to zero (in the standard topology on the reals) with this definition. (Contributed by NM, 7-Sep-2006.)
|- ~~> t = ( j e. Top |-> { <. f , x >. | ( f e. ( U. j ^pm CC ) /\ x e. U. j /\ A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
Theorem	lmrcl 12950	Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
|- ( F ( ~~> t ` J ) P -> J e. Top )
Theorem	lmfval 12951*	The relation "sequence f converges to point y " in a metric space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( J e. (TopOn ` X ) -> ( ~~> t ` J ) = { <. f , x >. | ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. J ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
Theorem	lmreltop 12952	The topological space convergence relation is a relation. (Contributed by Jim Kingdon, 25-Mar-2023.)
|- ( J e. Top -> Rel ( ~~> t ` J ) )
Theorem	cnfval 12953*	The set of all continuous functions from topology J to topology K. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( J Cn K ) = { f e. ( Y ^m X ) | A. y e. K ( `' f " y ) e. J } )
Theorem	cnpfval 12954*	The function mapping the points in a topology J to the set of all functions from J to topology K continuous at that point. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( J CnP K ) = ( x e. X |-> { f e. ( Y ^m X ) | A. w e. K ( ( f ` x ) e. w -> E. v e. J ( x e. v /\ ( f " v ) C_ w ) ) } ) )
Theorem	cnovex 12955	The class of all continuous functions from a topology to another is a set. (Contributed by Jim Kingdon, 14-Dec-2023.)
|- ( ( J e. Top /\ K e. Top ) -> ( J Cn K ) e. _V )
Theorem	iscn 12956*	The predicate "the class F is a continuous function from topology J to topology K". Definition of continuous function in [Munkres] p. 102. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. y e. K ( `' F " y ) e. J ) ) )
Theorem	cnpval 12957*	The set of all functions from topology J to topology K that are continuous at a point P. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( ( J CnP K ) ` P ) = { f e. ( Y ^m X ) | A. y e. K ( ( f ` P ) e. y -> E. x e. J ( P e. x /\ ( f " x ) C_ y ) ) } )
Theorem	iscnp 12958*	The predicate "the class F is a continuous function from topology J to topology K at point P". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) ) )
Theorem	iscn2 12959*	The predicate "the class F is a continuous function from topology J to topology K". Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   &    |- Y = U. K   =>    |- ( F e. ( J Cn K ) <-> ( ( J e. Top /\ K e. Top ) /\ ( F : X --> Y /\ A. y e. K ( `' F " y ) e. J ) ) )
Theorem	cntop1 12960	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( F e. ( J Cn K ) -> J e. Top )
Theorem	cntop2 12961	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( F e. ( J Cn K ) -> K e. Top )
Theorem	iscnp3 12962*	The predicate "the class F is a continuous function from topology J to topology K at point P". (Contributed by NM, 15-May-2007.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ x C_ ( `' F " y ) ) ) ) ) )
Theorem	cnf 12963	A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   &    |- Y = U. K   =>    |- ( F e. ( J Cn K ) -> F : X --> Y )
Theorem	cnf2 12964	A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Cn K ) ) -> F : X --> Y )
Theorem	cnprcl2k 12965	Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
|- ( ( J e. (TopOn ` X ) /\ K e. Top /\ F e. ( ( J CnP K ) ` P ) ) -> P e. X )
Theorem	cnpf2 12966	A continuous function at point P is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` P ) ) -> F : X --> Y )
Theorem	tgcn 12967*	The continuity predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K = ( topGen ` B ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   =>    |- ( ph -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. y e. B ( `' F " y ) e. J ) ) )
Theorem	tgcnp 12968*	The "continuous at a point" predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K = ( topGen ` B ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> P e. X )   =>    |- ( ph -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. B ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) ) )
Theorem	ssidcn 12969	The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) -> ( ( _I |` X ) e. ( J Cn K ) <-> K C_ J ) )
Theorem	icnpimaex 12970*	Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.)
|- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) )
Theorem	idcn 12971	A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
|- ( J e. (TopOn ` X ) -> ( _I |` X ) e. ( J Cn J ) )
Theorem	lmbr 12972*	Express the binary relation "sequence F converges to point P " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition F C_ ( CC X. X ) allows us to use objects more general than sequences when convenient; see the comment in df-lm 12949. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- ( ph -> J e. (TopOn ` X ) )   =>    |- ( ph -> ( F ( ~~> t ` J ) P <-> ( F e. ( X ^pm CC ) /\ P e. X /\ A. u e. J ( P e. u -> E. y e. ran ZZ>= ( F |` y ) : y --> u ) ) ) )
Theorem	lmbr2 12973*	Express the binary relation "sequence F converges to point P " in a metric space using an arbitrary upper set of integers. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   =>    |- ( ph -> ( F ( ~~> t ` J ) P <-> ( F e. ( X ^pm CC ) /\ P e. X /\ A. u e. J ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) ) ) )
Theorem	lmbrf 12974*	Express the binary relation "sequence F converges to point P " in a metric space using an arbitrary upper set of integers. This version of lmbr2 12973 presupposes that F is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F : Z --> X )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   =>    |- ( ph -> ( F ( ~~> t ` J ) P <-> ( P e. X /\ A. u e. J ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) A e. u ) ) ) )
Theorem	lmconst 12975	A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
|- Z = ( ZZ>= ` M )   =>    |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( Z X. { P } ) ( ~~> t ` J ) P )
Theorem	lmcvg 12976*	Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> P e. U )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ( ~~> t ` J ) P )   &    |- ( ph -> U e. J )   =>    |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( F ` k ) e. U )
Theorem	iscnp4 12977*	The predicate "the class F is a continuous function from topology J to topology K at point P " in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. ( ( nei ` K ) ` { ( F ` P ) } ) E. x e. ( ( nei ` J ) ` { P } ) ( F " x ) C_ y ) ) )
Theorem	cnpnei 12978*	A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top /\ F : X --> Y ) /\ A e. X ) -> ( F e. ( ( J CnP K ) ` A ) <-> A. y e. ( ( nei ` K ) ` { ( F ` A ) } ) ( `' F " y ) e. ( ( nei ` J ) ` { A } ) ) )
Theorem	cnima 12979	An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)
|- ( ( F e. ( J Cn K ) /\ A e. K ) -> ( `' F " A ) e. J )
Theorem	cnco 12980	The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. F ) e. ( J Cn L ) )
Theorem	cnptopco 12981	The composition of a function F continuous at P with a function continuous at ( F ` P ) is continuous at P. Proposition 2 of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
|- ( ( ( J e. Top /\ K e. Top /\ L e. Top ) /\ ( F e. ( ( J CnP K ) ` P ) /\ G e. ( ( K CnP L ) ` ( F ` P ) ) ) ) -> ( G o. F ) e. ( ( J CnP L ) ` P ) )
Theorem	cnclima 12982	A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( `' F " A ) e. ( Clsd ` J ) )
Theorem	cnntri 12983	Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- Y = U. K   =>    |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( ( int ` J ) ` ( `' F " S ) ) )
Theorem	cnntr 12984*	Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. x e. ~P Y ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) ) )
Theorem	cnss1 12985	If the topology K is finer than J, then there are more continuous functions from K than from J. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   =>    |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( J Cn L ) C_ ( K Cn L ) )
Theorem	cnss2 12986	If the topology K is finer than J, then there are fewer continuous functions into K than into J from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- Y = U. K   =>    |- ( ( L e. (TopOn ` Y ) /\ L C_ K ) -> ( J Cn K ) C_ ( J Cn L ) )
Theorem	cncnpi 12987	A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Cn K ) /\ A e. X ) -> F e. ( ( J CnP K ) ` A ) )
Theorem	cnsscnp 12988	The set of continuous functions is a subset of the set of continuous functions at a point. (Contributed by Raph Levien, 21-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   =>    |- ( P e. X -> ( J Cn K ) C_ ( ( J CnP K ) ` P ) )
Theorem	cncnp 12989*	A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 15-May-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. x e. X F e. ( ( J CnP K ) ` x ) ) ) )
Theorem	cncnp2m 12990*	A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Jim Kingdon, 30-Mar-2023.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( J e. Top /\ K e. Top /\ E. y y e. X ) -> ( F e. ( J Cn K ) <-> A. x e. X F e. ( ( J CnP K ) ` x ) ) )
Theorem	cnnei 12991*	Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( J e. Top /\ K e. Top /\ F : X --> Y ) -> ( F e. ( J Cn K ) <-> A. p e. X A. w e. ( ( nei ` K ) ` { ( F ` p ) } ) E. v e. ( ( nei ` J ) ` { p } ) ( F " v ) C_ w ) )
Theorem	cnconst2 12992	A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) -> ( X X. { B } ) e. ( J Cn K ) )
Theorem	cnconst 12993	A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
|- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( B e. Y /\ F : X --> { B } ) ) -> F e. ( J Cn K ) )
Theorem	cnrest 12994	Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
Theorem	cnrest2 12995	Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
|- ( ( K e. (TopOn ` Y ) /\ ran F C_ B /\ B C_ Y ) -> ( F e. ( J Cn K ) <-> F e. ( J Cn ( K ↾t B ) ) ) )
Theorem	cnrest2r 12996	Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
|- ( K e. Top -> ( J Cn ( K ↾t B ) ) C_ ( J Cn K ) )
Theorem	cnptopresti 12997	One direction of cnptoprest 12998 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 31-Mar-2023.)
|- ( ( ( J e. (TopOn ` X ) /\ K e. Top ) /\ ( A C_ X /\ P e. A /\ F e. ( ( J CnP K ) ` P ) ) ) -> ( F |` A ) e. ( ( ( J ↾t A ) CnP K ) ` P ) )
Theorem	cnptoprest 12998	Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 5-Apr-2023.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top /\ A C_ X ) /\ ( P e. ( ( int ` J ) ` A ) /\ F : X --> Y ) ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F |` A ) e. ( ( ( J ↾t A ) CnP K ) ` P ) ) )
Theorem	cnptoprest2 12999	Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : X --> B /\ B C_ Y ) ) -> ( F e. ( ( J CnP K ) ` P ) <-> F e. ( ( J CnP ( K ↾t B ) ) ` P ) ) )
Theorem	cndis 13000	Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( ~P A Cn J ) = ( X ^m A ) )