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
Syntax	clm 14901	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 14902*	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 14911 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 14903*	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 14904*	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 14905	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 14906	Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
|- ( F ( ~~> t ` J ) P -> J e. Top )
Theorem	lmfval 14907*	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 14908*	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 14909*	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 14910	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 14911*	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 14912*	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 14913*	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 14914*	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 14915	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( F e. ( J Cn K ) -> J e. Top )
Theorem	cntop2 14916	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( F e. ( J Cn K ) -> K e. Top )
Theorem	iscnp3 14917*	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 14918	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 14919	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 14920	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 14921	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 14922*	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 14923*	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 14924	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 14925*	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 14926	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 14927*	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 14904. (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 14928*	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 14929*	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 14928 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 14930	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 14931*	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 14932*	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 14933*	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 14934	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 14935	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 14936	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 14937	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 14938	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 14939*	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 14940	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 14941	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 14942	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 14943	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 14944*	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 14945*	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 14946*	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 14947	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 14948	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 14949	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 14950	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 14951	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 14952	One direction of cnptoprest 14953 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 14953	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 14954	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 14955	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 14956	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 14957	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 14958	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 14959	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 14960	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 14961	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 14962	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 14963*	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 14964	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 14965	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 14966	Extend class notation with the binary topological product operation.
class tX
Definition	df-tx 14967*	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 14968	Existence of the binary topological product. If R and S are known to be topologies, see txtop 14974. (Contributed by Jim Kingdon, 3-Aug-2023.)
|- ( ( R e. V /\ S e. W ) -> ( R tX S ) e. _V )
Theorem	txval 14969*	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 14970*	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 14971*	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 14972*	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 14973*	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 14974	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 14975	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 14976	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 14977	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 14978	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 14979	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 14980	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 14981	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 14982	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 14983	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 14984	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 14985*	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 14986*	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 14987*	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 14988*	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 14989	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 14990	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 14991	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 ) )
Theorem	txdis1cn 14992*	A function is jointly continuous on a discrete left topology iff it is continuous as a function of its right argument, for each fixed left value. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- ( ph -> X e. V )   &    |- ( ph -> J e. (TopOn ` Y ) )   &    |- ( ph -> K e. Top )   &    |- ( ph -> F Fn ( X X. Y ) )   &    |- ( ( ph /\ x e. X ) -> ( y e. Y |-> ( x F y ) ) e. ( J Cn K ) )   =>    |- ( ph -> F e. ( ( ~P X tX J ) Cn K ) )
Theorem	txlm 14993*	Two sequences converge iff the sequence of their ordered pairs converges. Proposition 14-2.6 of [Gleason] p. 230. (Contributed by NM, 16-Jul-2007.) (Revised by Mario Carneiro, 5-May-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> F : Z --> X )   &    |- ( ph -> G : Z --> Y )   &    |- H = ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. )   =>    |- ( ph -> ( ( F ( ~~> t ` J ) R /\ G ( ~~> t ` K ) S ) <-> H ( ~~> t ` ( J tX K ) ) <. R , S >. ) )
Theorem	lmcn2 14994*	The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 15-May-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> F : Z --> X )   &    |- ( ph -> G : Z --> Y )   &    |- ( ph -> F ( ~~> t ` J ) R )   &    |- ( ph -> G ( ~~> t ` K ) S )   &    |- ( ph -> O e. ( ( J tX K ) Cn N ) )   &    |- H = ( n e. Z |-> ( ( F ` n ) O ( G ` n ) ) )   =>    |- ( ph -> H ( ~~> t ` N ) ( R O S ) )
9.1.9  Continuous function-builders
Theorem	cnmptid 14995*	The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   =>    |- ( ph -> ( x e. X |-> x ) e. ( J Cn J ) )
Theorem	cnmptc 14996*	A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> P e. Y )   =>    |- ( ph -> ( x e. X |-> P ) e. ( J Cn K ) )
Theorem	cnmpt11 14997*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> ( x e. X |-> A ) e. ( J Cn K ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> ( y e. Y |-> B ) e. ( K Cn L ) )   &    |- ( y = A -> B = C )   =>    |- ( ph -> ( x e. X |-> C ) e. ( J Cn L ) )
Theorem	cnmpt11f 14998*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> ( x e. X |-> A ) e. ( J Cn K ) )   &    |- ( ph -> F e. ( K Cn L ) )   =>    |- ( ph -> ( x e. X |-> ( F ` A ) ) e. ( J Cn L ) )
Theorem	cnmpt1t 14999*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> ( x e. X |-> A ) e. ( J Cn K ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( J Cn L ) )   =>    |- ( ph -> ( x e. X |-> <. A , B >. ) e. ( J Cn ( K tX L ) ) )
Theorem	cnmpt12f 15000*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> ( x e. X |-> A ) e. ( J Cn K ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( J Cn L ) )   &    |- ( ph -> F e. ( ( K tX L ) Cn M ) )   =>    |- ( ph -> ( x e. X |-> ( A F B ) ) e. ( J Cn M ) )