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	resttopon2 14901	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 14902	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 14903	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 14904	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 14905	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 14906	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 14907	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 14908	Extend class notation with the class of continuous functions between topologies.
class Cn
Syntax	ccnp 14909	Extend class notation with the class of functions between topologies continuous at a given point.
class CnP
Syntax	clm 14910	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 14911*	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 14920 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 14912*	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 14913*	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 14914	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 14915	Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
|- ( F ( ~~> t ` J ) P -> J e. Top )
Theorem	lmfval 14916*	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 14917*	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 14918*	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 14919	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 14920*	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 14921*	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 14922*	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 14923*	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 14924	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( F e. ( J Cn K ) -> J e. Top )
Theorem	cntop2 14925	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( F e. ( J Cn K ) -> K e. Top )
Theorem	iscnp3 14926*	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 14927	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 14928	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 14929	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 14930	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 14931*	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 14932*	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 14933	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 14934*	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 14935	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 14936*	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 14913. (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 14937*	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 14938*	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 14937 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 14939	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 14940*	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 14941*	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 14942*	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 14943	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 14944	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 14945	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 14946	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 14947	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 14948*	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 14949	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 14950	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 14951	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 14952	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 14953*	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 14954*	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 14955*	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 14956	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 14957	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 14958	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 14959	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 14960	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 14961	One direction of cnptoprest 14962 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 14962	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 14963	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 14964	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 14965	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 14966	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 14967	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 14968	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 14969	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 14970	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 14971	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 14972*	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 )
Theorem	lmtopcnp 14973	The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
|- ( ph -> F ( ~~> t ` J ) P )   &    |- ( ph -> K e. Top )   &    |- ( ph -> G e. ( ( J CnP K ) ` P ) )   =>    |- ( ph -> ( G o. F ) ( ~~> t ` K ) ( G ` P ) )
Theorem	lmcn 14974	The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)
|- ( ph -> F ( ~~> t ` J ) P )   &    |- ( ph -> G e. ( J Cn K ) )   =>    |- ( ph -> ( G o. F ) ( ~~> t ` K ) ( G ` P ) )
9.1.8  Product topologies
Syntax	ctx 14975	Extend class notation with the binary topological product operation.
class tX
Definition	df-tx 14976*	Define the binary topological product, which is homeomorphic to the general topological product over a two element set, but is more convenient to use. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- tX = ( r e. _V , s e. _V |-> ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) )
Theorem	txvalex 14977	Existence of the binary topological product. If R and S are known to be topologies, see txtop 14983. (Contributed by Jim Kingdon, 3-Aug-2023.)
|- ( ( R e. V /\ S e. W ) -> ( R tX S ) e. _V )
Theorem	txval 14978*	Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)
|- B = ran ( x e. R , y e. S |-> ( x X. y ) )   =>    |- ( ( R e. V /\ S e. W ) -> ( R tX S ) = ( topGen ` B ) )
Theorem	txuni2 14979*	The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- B = ran ( x e. R , y e. S |-> ( x X. y ) )   &    |- X = U. R   &    |- Y = U. S   =>    |- ( X X. Y ) = U. B
Theorem	txbasex 14980*	The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- B = ran ( x e. R , y e. S |-> ( x X. y ) )   =>    |- ( ( R e. V /\ S e. W ) -> B e. _V )
Theorem	txbas 14981*	The set of Cartesian products of elements from two topological bases is a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- B = ran ( x e. R , y e. S |-> ( x X. y ) )   =>    |- ( ( R e. TopBases /\ S e. TopBases ) -> B e. TopBases )
Theorem	eltx 14982*	A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.)
|- ( ( J e. V /\ K e. W ) -> ( S e. ( J tX K ) <-> A. p e. S E. x e. J E. y e. K ( p e. ( x X. y ) /\ ( x X. y ) C_ S ) ) )
Theorem	txtop 14983	The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( R e. Top /\ S e. Top ) -> ( R tX S ) e. Top )
Theorem	txtopi 14984	The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)
|- R e. Top   &    |- S e. Top   =>    |- ( R tX S ) e. Top
Theorem	txtopon 14985	The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
|- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. (TopOn ` ( X X. Y ) ) )
Theorem	txuni 14986	The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- X = U. R   &    |- Y = U. S   =>    |- ( ( R e. Top /\ S e. Top ) -> ( X X. Y ) = U. ( R tX S ) )
Theorem	txunii 14987	The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)
|- R e. Top   &    |- S e. Top   &    |- X = U. R   &    |- Y = U. S   =>    |- ( X X. Y ) = U. ( R tX S )
Theorem	txopn 14988	The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( ( R e. V /\ S e. W ) /\ ( A e. R /\ B e. S ) ) -> ( A X. B ) e. ( R tX S ) )
Theorem	txss12 14989	Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- ( ( ( B e. V /\ D e. W ) /\ ( A C_ B /\ C C_ D ) ) -> ( A tX C ) C_ ( B tX D ) )
Theorem	txbasval 14990	It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)
|- ( ( R e. V /\ S e. W ) -> ( ( topGen ` R ) tX ( topGen ` S ) ) = ( R tX S ) )
Theorem	neitx 14991	The Cartesian product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top ) /\ ( A e. ( ( nei ` J ) ` C ) /\ B e. ( ( nei ` K ) ` D ) ) ) -> ( A X. B ) e. ( ( nei ` ( J tX K ) ) ` ( C X. D ) ) )
Theorem	tx1cn 14992	Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
|- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) )
Theorem	tx2cn 14993	Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
|- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) )
Theorem	txcnp 14994*	If two functions are continuous at D, then the ordered pair of them is continuous at D into the product topology. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> L e. (TopOn ` Z ) )   &    |- ( ph -> D e. X )   &    |- ( ph -> ( x e. X |-> A ) e. ( ( J CnP K ) ` D ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( ( J CnP L ) ` D ) )   =>    |- ( ph -> ( x e. X |-> <. A , B >. ) e. ( ( J CnP ( K tX L ) ) ` D ) )
Theorem	upxp 14995*	Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
|- P = ( 1st |` ( B X. C ) )   &    |- Q = ( 2nd |` ( B X. C ) )   =>    |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> E! h ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) )
Theorem	txcnmpt 14996*	A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- W = U. U   &    |- H = ( x e. W |-> <. ( F ` x ) , ( G ` x ) >. )   =>    |- ( ( F e. ( U Cn R ) /\ G e. ( U Cn S ) ) -> H e. ( U Cn ( R tX S ) ) )
Theorem	uptx 14997*	Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
|- T = ( R tX S )   &    |- X = U. R   &    |- Y = U. S   &    |- Z = ( X X. Y )   &    |- P = ( 1st |` Z )   &    |- Q = ( 2nd |` Z )   =>    |- ( ( F e. ( U Cn R ) /\ G e. ( U Cn S ) ) -> E! h e. ( U Cn T ) ( F = ( P o. h ) /\ G = ( Q o. h ) ) )
Theorem	txcn 14998	A map into the product of two topological spaces is continuous iff both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
|- X = U. R   &    |- Y = U. S   &    |- Z = ( X X. Y )   &    |- W = U. U   &    |- P = ( 1st |` Z )   &    |- Q = ( 2nd |` Z )   =>    |- ( ( R e. Top /\ S e. Top /\ F : W --> Z ) -> ( F e. ( U Cn ( R tX S ) ) <-> ( ( P o. F ) e. ( U Cn R ) /\ ( Q o. F ) e. ( U Cn S ) ) ) )
Theorem	txrest 14999	The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
|- ( ( ( R e. V /\ S e. W ) /\ ( A e. X /\ B e. Y ) ) -> ( ( R tX S ) ↾t ( A X. B ) ) = ( ( R ↾t A ) tX ( S ↾t B ) ) )
Theorem	txdis 15000	The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( ( A e. V /\ B e. W ) -> ( ~P A tX ~P B ) = ~P ( A X. B ) )