P. 150 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14901-15000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	neiss 14901	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 14902	A set is included in any of its neighborhoods. Generalization to subsets of elnei 14903. (Contributed by FL, 16-Nov-2006.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ N )
Theorem	elnei 14903	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 14904	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 14905*	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 14906	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 14907	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 14908	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 14909	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 14910	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 14911	The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 14909. (Contributed by FL, 2-Oct-2006.)
|- X = U. J   =>    |- ( J e. Top -> ( S C_ X <-> X e. ( ( nei ` J ) ` S ) ) )
Theorem	neiuni 14912	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 14913	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 14914	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 14915	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 14916*	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 14917	The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
|- ( J e. Top -> (/) e. ( ( nei ` J ) ` (/) ) )
9.1.6  Subspace topologies
Theorem	restrcl 14918	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 14919	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 14920	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 14921	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 14922	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 14923	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 14924	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 14925	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 14926	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 14927	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 14928	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 14929	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 14930	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 14931	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 14932	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 14933	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 14934	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 14935	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 )
9.1.7  Limits and continuity in topological spaces
Syntax	ccn 14936	Extend class notation with the class of continuous functions between topologies.
class Cn
Syntax	ccnp 14937	Extend class notation with the class of functions between topologies continuous at a given point.
class CnP
Syntax	clm 14938	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 14939*	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 14948 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 14940*	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 14941*	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	lmrel 14942	The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
|- Rel ( ~~> t ` J )
Theorem	lmrcl 14943	Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
|- ( F ( ~~> t ` J ) P -> J e. Top )
Theorem	lmfval 14944*	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	cnfval 14945*	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 14946*	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 14947	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 14948*	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 14949*	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 14950*	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 14951*	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 14952	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( F e. ( J Cn K ) -> J e. Top )
Theorem	cntop2 14953	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( F e. ( J Cn K ) -> K e. Top )
Theorem	iscnp3 14954*	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 14955	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 14956	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 14957	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 14958	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 14959*	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 14960*	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 14961	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 14962*	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 14963	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 14964*	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 14941. (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 14965*	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 14966*	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 14965 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 14967	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 14968*	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 14969*	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 14970*	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 14971	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 14972	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 14973	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 14974	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 14975	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 14976*	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 14977	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 14978	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 14979	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 14980	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 14981*	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 14982*	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 14983*	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 14984	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 14985	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 14986	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 14987	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 14988	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 14989	One direction of cnptoprest 14990 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 14990	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 14991	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 14992	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 ) )
Theorem	cnpdis 14993	If A is an isolated point in X (or equivalently, the singleton { A } is open in X), then every function is continuous at A. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> ( ( J CnP K ) ` A ) = ( Y ^m X ) )
Theorem	lmfpm 14994	If F converges, then F is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
|- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> F e. ( X ^pm CC ) )
Theorem	lmfss 14995	Inclusion of a function having a limit (used to ensure the limit relation is a set, under our definition). (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
|- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> F C_ ( CC X. X ) )
Theorem	lmcl 14996	Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
|- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> P e. X )
Theorem	lmss 14997	Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
|- K = ( J ↾t Y )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> Y e. V )   &    |- ( ph -> J e. Top )   &    |- ( ph -> P e. Y )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F : Z --> Y )   =>    |- ( ph -> ( F ( ~~> t ` J ) P <-> F ( ~~> t ` K ) P ) )
Theorem	sslm 14998	A finer topology has fewer convergent sequences (but the sequences that do converge, converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( ~~> t ` K ) C_ ( ~~> t ` J ) )
Theorem	lmres 14999	A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> F e. ( X ^pm CC ) )   &    |- ( ph -> M e. ZZ )   =>    |- ( ph -> ( F ( ~~> t ` J ) P <-> ( F |` ( ZZ>= ` M ) ) ( ~~> t ` J ) P ) )
Theorem	lmff 15000*	If F converges, there is some upper integer set on which F is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. dom ( ~~> t ` J ) )   =>    |- ( ph -> E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X )