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Theorem List for Metamath Proof Explorer - 26001-26100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexp58 26001 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ph  /\  ps )  /\  ( ( ch 
 /\  th )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp510 26002 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ( ps  /\  ch )  /\  th )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp511 26003 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ps  /\  ( ch 
 /\  th ) )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp512 26004 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ps  /\  ch )  /\  ( th  /\  ta ) ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theorem3com12d 26005 Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ph  ->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ( ch  /\  ps  /\  th ) )
 
Theoremimp5p 26006 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch  /\  th 
 /\  ta )  ->  et )
 ) )
 
Theoremimp5q 26007 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( ch 
 /\  th  /\  ta )  ->  et ) )
 
Theoremecase13d 26008 Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  ( ch  \/  ps 
 \/  th ) )   =>    |-  ( ph  ->  ps )
 
Theoremsubtr 26009 Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x Y   &    |-  F/_ x Z   &    |-  ( x  =  A  ->  X  =  Y )   &    |-  ( x  =  B  ->  X  =  Z )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  Y  =  Z ) )
 
Theoremsubtr2 26010 Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/ x ps   &    |-  F/ x ch   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ch ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  ( ps  <->  ch ) ) )
 
Theoremtrer 26011* A relation intersected with its converse is an equivalence relation if the relation is transitive. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( A. a A. b A. c ( ( a 
 .<_  b  /\  b  .<_  c )  ->  a  .<_  c )  ->  (  .<_  i^i  `'  .<_  )  Er  dom  (  .<_  i^i  `'  .<_  ) )
 
Theoremelicc3 26012 An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) ) )
 
Theoremfinminlem 26013* A useful lemma about finite sets. If a property holds for a finite set, it holds for a minimal set. (Contributed by Jeff Hankins, 4-Dec-2009.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  Fin  ph  ->  E. x ( ph  /\  A. y
 ( ( y  C_  x  /\  ps )  ->  x  =  y )
 ) )
 
Theoremgtinf 26014* Any number greater than an infimum is greater than some element of the set. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  (
 ( ( S  C_  RR  /\  S  =/=  (/)  /\  E. x  e.  RR  A. y  e.  S  x  <_  y
 )  /\  ( A  e.  RR  /\  sup ( S ,  RR ,  `'  <  )  <  A ) )  ->  E. z  e.  S  z  <  A )
 
Theoremopnrebl 26015* A set is open in the standard topology of the reals precisely when every point can be enclosed in an open ball. (Contributed by Jeff Hankins, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
 |-  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. x  e.  A  E. y  e.  RR+  ( ( x  -  y ) (,) ( x  +  y )
 )  C_  A )
 )
 
Theoremopnrebl2 26016* A set is open in the standard topology of the reals precisely when every point can be enclosed in an arbitrarily small ball. (Contributed by Jeff Hankins, 22-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
 |-  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  ( z  <_  y  /\  ( ( x  -  z ) (,) ( x  +  z )
 )  C_  A )
 ) )
 
Theoremnn0prpwlem 26017* Lemma for nn0prpw 26018. Use strong induction to show that every natural number has unique prime power divisors. (Contributed by Jeff Hankins, 28-Sep-2013.)
 |-  ( A  e.  NN  ->  A. k  e.  NN  (
 k  <  A  ->  E. p  e.  Prime  E. n  e.  NN  -.  ( ( p ^ n ) 
 ||  k  <->  ( p ^ n )  ||  A ) ) )
 
Theoremnn0prpw 26018* Two nonnegative integers are the same if and only if they are divisible by the same prime powers. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  =  B  <->  A. p  e.  Prime  A. n  e.  NN  ( ( p ^ n )  ||  A 
 <->  ( p ^ n )  ||  B ) ) )
 
19.13.2  Basic topological facts
 
Theoremtopbnd 26019 Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( ( ( cls `  J ) `  A )  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J ) `  A )  \  ( ( int `  J ) `  A ) ) )
 
Theoremopnbnd 26020 A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( A  e.  J  <->  ( A  i^i  ( ( ( cls `  J ) `  A )  i^i  ( ( cls `  J ) `  ( X  \  A ) ) ) )  =  (/) ) )
 
Theoremcldbnd 26021 A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  ( ( ( cls `  J ) `  A )  i^i  (
 ( cls `  J ) `  ( X  \  A ) ) )  C_  A ) )
 
Theoremntruni 26022* A union of interiors is a subset of the interior of the union. The reverse inclusion may not hold. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  O  C_  ~P X )  ->  U_ o  e.  O  ( ( int `  J ) `  o )  C_  ( ( int `  J ) `  U. O ) )
 
Theoremclsun 26023 A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  ( ( cls `  J ) `  ( A  u.  B ) )  =  ( ( ( cls `  J ) `  A )  u.  ( ( cls `  J ) `  B ) ) )
 
Theoremclsint2 26024* The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  ( ( cls `  J ) `  |^| C )  C_  |^|_ c  e.  C  ( ( cls `  J ) `  c ) )
 
Theoremopnregcld 26025* A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( ( ( cls `  J ) `  (
 ( int `  J ) `  A ) )  =  A  <->  E. o  e.  J  A  =  ( ( cls `  J ) `  o ) ) )
 
Theoremcldregopn 26026* A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( ( ( int `  J ) `  (
 ( cls `  J ) `  A ) )  =  A  <->  E. c  e.  ( Clsd `  J ) A  =  ( ( int `  J ) `  c
 ) ) )
 
Theoremneiin 26027 Two neighborhoods intersect to form a neighborhood of the intersection. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  (
 ( J  e.  Top  /\  M  e.  ( ( nei `  J ) `  A )  /\  N  e.  ( ( nei `  J ) `  B ) ) 
 ->  ( M  i^i  N )  e.  ( ( nei `  J ) `  ( A  i^i  B ) ) )
 
Theoremhmeoclda 26028 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
 |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Homeo  K ) )  /\  S  e.  ( Clsd `  J ) )  ->  ( F " S )  e.  ( Clsd `  K ) )
 
Theoremhmeocldb 26029 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.)
 |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Homeo  K ) )  /\  S  e.  ( Clsd `  K ) )  ->  ( `' F " S )  e.  ( Clsd `  J ) )
 
19.13.3  Topology of the real numbers
 
TheoremivthALT 26030* An alternate proof of the Intermediate Value Theorem ivth 19219 using topology. (Contributed by Jeff Hankins, 17-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn->
 CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A ) (,) ( F `  B ) ) ) ) )  ->  E. x  e.  ( A (,) B ) ( F `  x )  =  U )
 
19.13.4  Refinements
 
Syntaxcfne 26031 Extend class definition to include the "finer than" relation.
 class  Fne
 
Syntaxcref 26032 Extend class definition to include the refinement relation.
 class  Ref
 
Syntaxcptfin 26033 Extend class definition to include the class of point-finite covers.
 class  PtFin
 
Syntaxclocfin 26034 Extend class definition to include the class of locally finite covers.
 class  LocFin
 
Definitiondf-fne 26035* Define the fineness relation for covers. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  Fne  =  { <. x ,  y >.  |  ( U. x  =  U. y  /\  A. z  e.  x  z  C_ 
 U. ( y  i^i 
 ~P z ) ) }
 
Definitiondf-ref 26036* Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  Ref  =  { <. x ,  y >.  |  ( U. x  =  U. y  /\  A. z  e.  y  E. w  e.  x  z  C_  w ) }
 
Definitiondf-ptfin 26037* Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  PtFin  =  { x  |  A. y  e. 
 U. x { z  e.  x  |  y  e.  z }  e.  Fin }
 
Definitiondf-locfin 26038* Define "locally finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  LocFin  =  ( x  e.  Top  |->  { y  |  ( U. x  = 
 U. y  /\  A. p  e.  U. x E. n  e.  x  ( p  e.  n  /\  { s  e.  y  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) } )
 
Theoremfnerel 26039 Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  Rel  Fne
 
Theoremisfne 26040* The predicate " B is finer than  A." This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x  C_ 
 U. ( B  i^i  ~P x ) ) ) )
 
Theoremisfne4 26041 The predicate " B is finer than  A " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B ) ) )
 
Theoremisfne4b 26042 A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  V  ->  ( A Fne B  <->  ( X  =  Y  /\  ( topGen `  A )  C_  ( topGen `  B )
 ) ) )
 
Theoremisfne2 26043* The predicate " B is finer than  A." (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  A. y  e.  x  E. z  e.  B  (
 y  e.  z  /\  z  C_  x ) ) ) )
 
Theoremisfne3 26044* The predicate " B is finer than  A." (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  E. y ( y  C_  B  /\  x  =  U. y ) ) ) )
 
Theoremfnebas 26045 A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( A Fne B  ->  X  =  Y )
 
Theoremfnetg 26046 A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( A Fne B  ->  A  C_  ( topGen `  B )
 )
 
Theoremfnessex 26047* If  B is finer than  A and  S is an element of  A, every point in  S is an element of a subset of  S which is in  B. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  (
 ( A Fne B  /\  S  e.  A  /\  P  e.  S )  ->  E. x  e.  B  ( P  e.  x  /\  x  C_  S ) )
 
Theoremfneuni 26048* If  B is finer than  A, every element of  A is a union of elements of  B. (Contributed by Jeff Hankins, 11-Oct-2009.)
 |-  (
 ( A Fne B  /\  S  e.  A ) 
 ->  E. x ( x 
 C_  B  /\  S  =  U. x ) )
 
Theoremfneint 26049* If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)
 |-  ( A Fne B  ->  |^| { x  e.  B  |  P  e.  x }  C_  |^| { x  e.  A  |  P  e.  x } )
 
Theoremrefrel 26050 Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  Rel  Ref
 
Theoremisref 26051* The property of being a refinement of a cover. Dr. Nyikos once commented in class that the term "refinement" is actually misleading and that people are inclined to confuse it with the notion defined in isfne 26040. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Ref B  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) ) )
 
Theoremrefbas 26052 A refinement covers the same set. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( A Ref B  ->  X  =  Y )
 
Theoremrefssex 26053* Every set in a refinement has a superset in the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  (
 ( A Ref B  /\  S  e.  B ) 
 ->  E. x  e.  A  S  C_  x )
 
Theoremfness 26054 A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( ( B  e.  C  /\  A  C_  B  /\  X  =  Y ) 
 ->  A Fne B )
 
Theoremssref 26055 A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y ) 
 ->  B Ref A )
 
Theoremfneref 26056 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
 |-  ( A  e.  V  ->  A Fne A )
 
Theoremrefref 26057 Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  ( A  e.  V  ->  A Ref A )
 
Theoremfnetr 26058 Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  (
 ( A Fne B  /\  B Fne C ) 
 ->  A Fne C )
 
Theoremfneval 26059 Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  .~  B 
 <->  ( topGen `  A )  =  ( topGen `  B )
 ) )
 
Theoremfneer 26060 Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |- 
 .~  Er  _V
 
Theoremreftr 26061 Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  (
 ( A Ref B  /\  B Ref C ) 
 ->  A Ref C )
 
Theoremtopfne 26062 Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( K  e.  Top  /\  X  =  Y ) 
 ->  ( J  C_  K  <->  J Fne K ) )
 
Theoremtopfneec 26063 A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |-  ( J  e.  Top  ->  ( A  e.  [ J ]  .~  <->  ( topGen `  A )  =  J )
 )
 
Theoremtopfneec2 26064 A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( [ J ]  .~  =  [ K ]  .~  <->  J  =  K ) )
 
Theoremfnessref 26065* A cover is finer iff it has a subcover which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( X  =  Y  ->  ( A Fne B  <->  E. c ( c  C_  B  /\  A ( Fne 
 i^i  Ref ) c ) ) )
 
Theoremrefssfne 26066* A cover is a refinement iff it is a subcover of something which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( X  =  Y  ->  ( A Ref B  <->  E. c ( B  C_  c  /\  A ( Fne 
 i^i  Ref ) c ) ) )
 
Theoremisptfin 26067* The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. A   =>    |-  ( A  e.  B  ->  ( A  e.  PtFin  <->  A. x  e.  X  { y  e.  A  |  x  e.  y }  e.  Fin ) )
 
Theoremislocfin 26068* The statement "is a locally finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. J   &    |-  Y  =  U. A   =>    |-  ( A  e.  ( LocFin `
  J )  <->  ( J  e.  Top  /\  X  =  Y  /\  A. x  e.  X  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
 
Theoremfinptfin 26069 A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  ( A  e.  Fin  ->  A  e.  PtFin )
 
Theoremptfinfin 26070* A point covered by a point-finite cover is only covered by finitely many elements. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. A   =>    |-  ( ( A  e.  PtFin  /\  P  e.  X ) 
 ->  { x  e.  A  |  P  e.  x }  e.  Fin )
 
Theoremfinlocfin 26071 A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. J   &    |-  Y  =  U. A   =>    |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  A  e.  ( LocFin `  J ) )
 
Theoremlocfintop 26072 A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  ( A  e.  ( LocFin `  J )  ->  J  e.  Top )
 
Theoremlocfinbas 26073 A locally finite cover must cover the base set of its corresponding topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. J   &    |-  Y  =  U. A   =>    |-  ( A  e.  ( LocFin `
  J )  ->  X  =  Y )
 
Theoremlocfinnei 26074* A point covered by a locally finite cover has a neighborhood which intersects only finitely many elements of the cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. J   =>    |-  ( ( A  e.  ( LocFin `  J )  /\  P  e.  X ) 
 ->  E. n  e.  J  ( P  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
 
Theoremlfinpfin 26075 A locally finite cover is point-finite. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  ( A  e.  ( LocFin `  J )  ->  A  e.  PtFin
 )
 
Theoremlocfincmp 26076 For a compact space, the locally finite covers are precisely the finite covers. Sadly, this property does not properly characterize all compact spaces. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. C   =>    |-  ( J  e.  Comp  ->  ( C  e.  ( LocFin `
  J )  <->  ( C  e.  Fin  /\  X  =  Y ) ) )
 
Theoremlocfindis 26077 The locally finite covers of a discrete space are precisely the point-finite covers. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  Y  =  U. C   =>    |-  ( C  e.  ( LocFin `
  ~P X )  <-> 
 ( C  e.  PtFin  /\  X  =  Y ) )
 
Theoremlocfincf 26078 A locally finite cover in a coarser topology is locally finite in a finer topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. J   =>    |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( LocFin `  J )  C_  ( LocFin `  K )
 )
 
Theoremcomppfsc 26079* A space where every open cover has a point-finite subcover is compact. This is significant in part because it shows half of the proposition that if only half the generalization in the definition of metacompactness (and consequently paracompactness) is performed, one does not obtain any more spaces. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( J  e.  Comp  <->  A. c  e.  ~P  J ( X  =  U. c  ->  E. d  e.  PtFin  ( d  C_  c  /\  X  =  U. d ) ) ) )
 
19.13.5  Neighborhood bases determine topologies
 
Theoremneibastop1 26080* A collection of neighborhood bases determines a topology. Part of Theorem 4.5 of Stephen Willard's General Topology. (Contributed by Jeff Hankins, 8-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( ~P ~P X  \  { (/) } )
 )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x )  /\  w  e.  ( F `  x ) ) )  ->  ( ( F `  x )  i^i  ~P (
 v  i^i  w )
 )  =/=  (/) )   &    |-  J  =  { o  e.  ~P X  |  A. x  e.  o  ( ( F `
  x )  i^i 
 ~P o )  =/=  (/) }   =>    |-  ( ph  ->  J  e.  (TopOn `  X )
 )
 
Theoremneibastop2lem 26081* Lemma for neibastop2 26082. (Contributed by Jeff Hankins, 12-Sep-2009.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( ~P ~P X  \  { (/) } )
 )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x )  /\  w  e.  ( F `  x ) ) )  ->  ( ( F `  x )  i^i  ~P (
 v  i^i  w )
 )  =/=  (/) )   &    |-  J  =  { o  e.  ~P X  |  A. x  e.  o  ( ( F `
  x )  i^i 
 ~P o )  =/=  (/) }   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  x  e.  v )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  E. t  e.  ( F `  x ) A. y  e.  t  ( ( F `  y )  i^i  ~P v
 )  =/=  (/) )   &    |-  ( ph  ->  P  e.  X )   &    |-  ( ph  ->  N  C_  X )   &    |-  ( ph  ->  U  e.  ( F `  P ) )   &    |-  ( ph  ->  U  C_  N )   &    |-  G  =  ( rec ( ( a  e. 
 _V  |->  U_ z  e.  a  U_ x  e.  X  ( ( F `  x )  i^i  ~P z ) ) ,  { U } )  |`  om )   &    |-  S  =  { y  e.  X  |  E. f  e.  U. ran  G ( ( F `
  y )  i^i 
 ~P f )  =/=  (/) }   =>    |-  ( ph  ->  E. u  e.  J  ( P  e.  u  /\  u  C_  N ) )
 
Theoremneibastop2 26082* In the topology generated by a neighborhood base, a set is a neighborhood of a point iff it contains a subset in the base. (Contributed by Jeff Hankins, 9-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( ~P ~P X  \  { (/) } )
 )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x )  /\  w  e.  ( F `  x ) ) )  ->  ( ( F `  x )  i^i  ~P (
 v  i^i  w )
 )  =/=  (/) )   &    |-  J  =  { o  e.  ~P X  |  A. x  e.  o  ( ( F `
  x )  i^i 
 ~P o )  =/=  (/) }   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  x  e.  v )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  E. t  e.  ( F `  x ) A. y  e.  t  ( ( F `  y )  i^i  ~P v
 )  =/=  (/) )   =>    |-  ( ( ph  /\  P  e.  X ) 
 ->  ( N  e.  (
 ( nei `  J ) `  { P } )  <->  ( N  C_  X  /\  ( ( F `  P )  i^i  ~P N )  =/=  (/) ) ) )
 
Theoremneibastop3 26083* The topology generated by a neighborhood base is unique. (Contributed by Jeff Hankins, 16-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F : X --> ( ~P ~P X  \  { (/) } )
 )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x )  /\  w  e.  ( F `  x ) ) )  ->  ( ( F `  x )  i^i  ~P (
 v  i^i  w )
 )  =/=  (/) )   &    |-  J  =  { o  e.  ~P X  |  A. x  e.  o  ( ( F `
  x )  i^i 
 ~P o )  =/=  (/) }   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  x  e.  v )   &    |-  ( ( ph  /\  ( x  e.  X  /\  v  e.  ( F `  x ) ) )  ->  E. t  e.  ( F `  x ) A. y  e.  t  ( ( F `  y )  i^i  ~P v
 )  =/=  (/) )   =>    |-  ( ph  ->  E! j  e.  (TopOn `  X ) A. x  e.  X  ( ( nei `  j ) `  { x } )  =  { n  e.  ~P X  |  ( ( F `  x )  i^i  ~P n )  =/=  (/) } )
 
19.13.6  Lattice structure of topologies
 
Theoremtopmtcl 26084 The meet of a collection of topologies on  X is again a topology on  X. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  S  C_  (TopOn `  X ) )  ->  ( ~P X  i^i  |^| S )  e.  (TopOn `  X ) )
 
Theoremtopmeet 26085* Two equivalent formulations of the meet of a collection of topologies. (Contributed by Jeff Hankins, 4-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  S  C_  (TopOn `  X ) )  ->  ( ~P X  i^i  |^| S )  =  U. { k  e.  (TopOn `  X )  |  A. j  e.  S  k  C_  j } )
 
Theoremtopjoin 26086* Two equivalent formulations of the join of a collection of topologies. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  S  C_  (TopOn `  X ) )  ->  ( topGen `  ( fi `  ( { X }  u.  U. S ) ) )  = 
 |^| { k  e.  (TopOn `  X )  |  A. j  e.  S  j  C_  k } )
 
Theoremfnemeet1 26087* The meet of a collection of equivalence classes of covers with respect to fineness. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  A. y  e.  S  X  =  U. y  /\  A  e.  S )  ->  ( ~P X  i^i  |^|_
 t  e.  S  (
 topGen `  t ) ) Fne A )
 
Theoremfnemeet2 26088* The meet of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  A. y  e.  S  X  =  U. y ) 
 ->  ( T Fne ( ~P X  i^i  |^|_ t  e.  S  ( topGen `  t
 ) )  <->  ( X  =  U. T  /\  A. x  e.  S  T Fne x ) ) )
 
Theoremfnejoin1 26089* Join of equivalence classes under the fineness relation-part one. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  A. y  e.  S  X  =  U. y  /\  A  e.  S )  ->  A Fne if ( S  =  (/) ,  { X } ,  U. S ) )
 
Theoremfnejoin2 26090* Join of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  (
 ( X  e.  V  /\  A. y  e.  S  X  =  U. y ) 
 ->  ( if ( S  =  (/) ,  { X } ,  U. S ) Fne T  <->  ( X  =  U. T  /\  A. x  e.  S  x Fne T ) ) )
 
19.13.7  Filter bases
 
Theoremfgmin 26091 Minimality property of a generated filter: every filter that contains  B contains its generated filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
 |-  (
 ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X ) )  ->  ( B  C_  F  <->  ( X filGen B )  C_  F )
 )
 
Theoremneifg 26092* The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 17796. (Contributed by Jeff Hankins, 3-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( X filGen { x  e.  J  |  S  C_  x } )  =  ( ( nei `  J ) `  S ) )
 
19.13.8  Directed sets, nets
 
Theoremtailfval 26093* The tail function for a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  X  =  dom  D   =>    |-  ( D  e.  DirRel  ->  ( tail `  D )  =  ( x  e.  X  |->  ( D " { x } ) ) )
 
Theoremtailval 26094 The tail of an element in a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  X  =  dom  D   =>    |-  ( ( D  e.  DirRel  /\  A  e.  X ) 
 ->  ( ( tail `  D ) `  A )  =  ( D " { A } ) )
 
Theoremeltail 26095 An element of a tail. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  X  =  dom  D   =>    |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  (
 ( tail `  D ) `  A )  <->  A D B ) )
 
Theoremtailf 26096 The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  X  =  dom  D   =>    |-  ( D  e.  DirRel  ->  ( tail `  D ) : X --> ~P X )
 
Theoremtailini 26097 A tail contains its initial element. (Contributed by Jeff Hankins, 25-Nov-2009.)
 |-  X  =  dom  D   =>    |-  ( ( D  e.  DirRel  /\  A  e.  X ) 
 ->  A  e.  ( (
 tail `  D ) `  A ) )
 
Theoremtailfb 26098 The collection of tails of a directed set is a filter base. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  X  =  dom  D   =>    |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ran  ( tail `  D )  e.  ( fBas `  X )
 )
 
Theoremfilnetlem1 26099* Lemma for filnet 26103. Change variables. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  H  =  U_ n  e.  F  ( { n }  X.  n )   &    |-  D  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  ( 1st `  y )  C_  ( 1st `  x )
 ) }   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A D B  <->  ( ( A  e.  H  /\  B  e.  H ) 
 /\  ( 1st `  B )  C_  ( 1st `  A ) ) )
 
Theoremfilnetlem2 26100* Lemma for filnet 26103. The field of the direction. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  H  =  U_ n  e.  F  ( { n }  X.  n )   &    |-  D  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  ( 1st `  y )  C_  ( 1st `  x )
 ) }   =>    |-  ( (  _I  |`  H ) 
 C_  D  /\  D  C_  ( H  X.  H ) )
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