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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtopfne 25701 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 25702 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 25703 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 25704* 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 25705* 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 25706* 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 25707* 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 25708 A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  ( A  e.  Fin  ->  A  e.  PtFin )
 
Theoremptfinfin 25709* 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 25710 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 25711 A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  ( A  e.  ( LocFin `  J )  ->  J  e.  Top )
 
Theoremlocfinbas 25712 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 25713* 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 25714 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 25715 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 25716 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 25717 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 25718* 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 ) ) ) )
 
18.14.5  Neighborhood bases determine topologies
 
Theoremneibastop1 25719* 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 25720* Lemma for neibastop2 25721. (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 25721* 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 25722* 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 )  =/=  (/) } )
 
18.14.6  Lattice structure of topologies
 
Theoremtopmtcl 25723 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 25724* 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 25725* 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 25726* 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 25727* 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 25728* 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 25729* 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 ) ) )
 
18.14.7  Filter bases
 
Theoremfgmin 25730 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 25731* The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 17533. (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 ) )
 
18.14.8  Directed sets, nets
 
Theoremtailfval 25732* 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 25733 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 25734 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 25735 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 25736 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 25737 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 25738* Lemma for filnet 25742. 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 25739* Lemma for filnet 25742. 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 ) )
 
Theoremfilnetlem3 25740* Lemma for filnet 25742. (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 )
 ) }   =>    |-  ( H  =  U. U. D  /\  ( F  e.  ( Fil `  X )  ->  ( H  C_  ( F  X.  X ) 
 /\  D  e.  DirRel ) ) )
 
Theoremfilnetlem4 25741* Lemma for filnet 25742. (Contributed by Jeff Hankins, 15-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 )
 ) }   =>    |-  ( F  e.  ( Fil `  X )  ->  E. d  e.  DirRel  E. f
 ( f :  dom  d
 --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  d ) ) ) )
 
Theoremfilnet 25742* A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  E. d  e.  DirRel  E. f ( f :  dom  d --> X  /\  F  =  ( ( X  FilMap  f ) `  ran  ( tail `  d )
 ) ) )
 
18.15  Mathbox for Jeff Madsen
 
18.15.1  Logic and set theory
 
Theoremanim12da 25743 Conjoin antecedents and consequents in a deduction. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( ph  /\  ps )  ->  th )   &    |-  ( ( ph  /\ 
 ch )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ch ) )  ->  ( th  /\  ta )
 )
 
Theorembiadan2OLD 25744 Add a conjunction to an equivalence. (Moved to biadan2 623 in main set.mm and may be deleted by mathbox owner, JM. --NM 25-Feb-2014.) (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ph  <->  ( ps  /\  ch ) )
 
Theoremsyldanl 25745 A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   &    |-  ( ( (
 ph  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ph  /\  ps )  /\  th )  ->  ta )
 
Theoremsyl3an2cOLDOLD 25746 A syllogism inference combined with contraction. (Moved to syl13anc 1184 in main set.mm and may be deleted by mathbox owner, JM. --NM 8-Sep-2011.) (Contributed by Jeff Madsen, 17-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ( ps 
 /\  ch  /\  th )
 )  ->  ta )   &    |-  ( et  ->  ph )   &    |-  ( et  ->  ps )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  th )   =>    |-  ( et  ->  ta )
 
Theoremmp4anOLD 25747 An inference based on modus ponens. (Moved to mp4an 654 in main set.mm and may be deleted by mathbox owner, JM. --NM 19-Oct-2012.) (Contributed by Jeff Madsen, 15-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ch   &    |-  th   &    |-  ( ( (
 ph  /\  ps )  /\  ( ch  /\  th ) )  ->  ta )   =>    |-  ta
 
TheoremimdistandaOLD 25748 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Moved into main set.mm as imdistanda 674 and may be deleted by mathbox owner, SF. --NM 20-Sep-2013.) (Contributed by Jeff Madsen, 19-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  ( ch  ->  th )
 )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  ( ps  /\  th ) ) )
 
Theoremr19.21aivvaOLD 25749* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Moved to ralrimivva 2636 in main set.mm and may be deleted by mathbox owner, JM. --NM 17-Jul-2012.) (Contributed by Jeff Madsen, 19-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  ps )   =>    |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )
 
TheoremsbmoOLD 25750* Substitution into "at most one". (Moved to sbmo 2174 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Jan-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
 
Theorem2ralorOLD 25751* Distribute quantification over "or". (Moved to 2ralor 2710 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Jan-2012.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  e.  A  A. y  e.  B  (
 ph  \/  ps )  <->  (
 A. x  e.  A  ph 
 \/  A. y  e.  B  ps ) )
 
TheoremrexunOLD 25752 Restricted existential quantification over union. (Moved to rexun 3356 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Jan-2012.) (Contributed by Jeff Madsen, 5-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x  e.  ( A  u.  B ) ph  <->  ( E. x  e.  A  ph 
 \/  E. x  e.  B  ph ) )
 
TheoremralunOLD 25753* Restricted quantification over union. (Moved to ralun 3358 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Jan-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A. x  e.  A  ph 
 /\  A. x  e.  B  ph )  ->  A. x  e.  ( A  u.  B ) ph )
 
TheoremrexsnOLD 25754* Restricted existential quantification over a singleton. (Moved to rexsn 3676 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Jan-2012.) (Contributed by Jeff Madsen, 5-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e. 
 { A } ph  <->  ps )
 
Theoremrexcom4aOLD 25755* Specialized existential commutation lemma. (Moved to rexcom4a 2809 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Apr-2012.) (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x E. y  e.  A  ( ph  /\  ps ) 
 <-> 
 E. y  e.  A  ( ph  /\  E. x ps ) )
 
Theoremrexcom4bOLD 25756* Specialized existential commutation lemma. (Moved to rexcom4b 2810 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Apr-2012.) (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  e.  _V   =>    |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B ) 
 <-> 
 E. y  e.  A  ph )
 
Theorem3reeanvOLD 25757* Rearrange three existential quantifiers. (Moved to 3reeanv 2709 in main set.mm and may be deleted by mathbox owner, JM. --NM 21-Mar-2013.) (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x  e.  A  E. y  e.  B  E. z  e.  C  ( ph  /\  ps  /\  ch )  <->  ( E. x  e.  A  ph  /\  E. y  e.  B  ps  /\  E. z  e.  C  ch ) )
 
TheoremmorexOLD 25758* Derive membership from uniqueness. (Moved to morex 2950 in main set.mm and may be deleted by mathbox owner, JM. --NM 19-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  e.  _V   &    |-  ( x  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( ( E. x  e.  A  ph  /\  E* x ph )  ->  ( ps  ->  B  e.  A ) )
 
Theoremeuuni2OLD 25759* The unique element such that 
ph. (Moved to iota2 6279 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( A  e.  B  /\  E! x ph )  ->  ( ps 
 <-> 
 U. { x  |  ph
 }  =  A ) )
 
Theoremunirep 25760* Define a quantity whose definition involves a choice of representative, but which is uniquely determined regardless of the choice. (Contributed by Jeff Madsen, 1-Jun-2011.)
 |-  (
 y  =  D  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  D  ->  B  =  C )   &    |-  (
 y  =  z  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  z  ->  B  =  F )   &    |-  B  e.  _V   =>    |-  ( ( A. y  e.  A  A. z  e.  A  ( ( ph  /\ 
 ch )  ->  B  =  F )  /\  ( D  e.  A  /\  ps ) )  ->  ( iota x E. y  e.  A  ( ph  /\  x  =  B ) )  =  C )
 
Theoremrabeq12OLD 25761* Equality of restricted class abstractions. (Moved to rabeqbidv 2784 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Sep-2011.) (Contributed by Jeff Madsen, 1-Dec-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch }
 )
 
Theoremrabeq0OLD 25762* Condition for a restricted class abstraction to be empty. (Moved to rabeq0 3477 in main set.mm and may be deleted by mathbox owner, JM. --NM 1-Apr-2013.) (Contributed by Jeff Madsen, 7-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( { x  e.  A  |  ph }  =  (/)  <->  A. x  e.  A  -.  ph )
 
TheoremrabxmOLD 25763* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) (Moved to rabxm 3478 in main set.mm and may be deleted by mathbox owner, JM. --NM 10-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  =  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  -.  ph } )
 
TheoremrabncOLD 25764* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) (Moved to rabnc 3479 in main set.mm and may be deleted by mathbox owner, JM. --NM 10-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  (/)
 
TheoremralabOLD 25765* Universal quantification over a class abstraction. (Moved to ralab 2927 in main set.mm and may be deleted by mathbox owner, JM. --NM 20-Oct-2011.) (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 y  =  x  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  { y  |  ph } ch  <->  A. x ( ps 
 ->  ch ) )
 
TheoremralrabOLD 25766* Universal quantification over a restricted class abstraction. (Moved to ralrab 2928 in main set.mm and may be deleted by mathbox owner, JM. --NM 20-Oct-2011.) (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 y  =  x  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  { y  e.  A  |  ph } ch  <->  A. x  e.  A  ( ps  ->  ch )
 )
 
TheoremrexrabOLD 25767* Existential quantification over a class abstraction. (Moved to rexrab 2930 in main set.mm and may be deleted by mathbox owner, JM. --NM 20-Mar-2013.) (Contributed by Jeff Madsen, 17-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. y  e.  { x  e.  A  |  ph } ch  <->  E. y  e.  A  ( ps  /\  ch )
 )
 
TheoremraleqfnOLD 25768* Change the domain of quantification by a function. (Moved to ralima 5720 in main set.mm and may be deleted by mathbox owner, JM. --NM 17-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 y  =  ( F `
  x )  ->  ( ph  <->  ps ) )   =>    |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( A. x  e.  B  ps 
 <-> 
 A. y  e.  ( F " B ) ph ) )
 
Theoremcover2 25769* Two ways of expressing the statement "there is a cover of  A by elements of  B such that for each set in the cover,  ph." Note that  ph and  x must be distinct. (Contributed by Jeff Madsen, 20-Jun-2010.)
 |-  B  e.  _V   &    |-  A  =  U. B   =>    |-  ( A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  ph )  <->  E. z  e.  ~P  B ( U. z  =  A  /\  A. y  e.  z  ph ) )
 
Theoremcover2g 25770* Two ways of expressing the statement "there is a cover of  A by elements of  B such that for each set in the cover,  ph." Note that  ph and  x must be distinct. (Contributed by Jeff Madsen, 21-Jun-2010.)
 |-  A  =  U. B   =>    |-  ( B  e.  C  ->  ( A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  ph )  <->  E. z  e.  ~P  B ( U. z  =  A  /\  A. y  e.  z  ph ) ) )
 
TheoremdisjrOLD 25771* Two ways of saying that two classes are disjoint. (Moved to disjr 3497 in main set.mm and may be deleted by mathbox owner, JM. --NM 23-Mar-2013.) (Contributed by Jeff Madsen, 19-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  i^i  B )  =  (/)  <->  A. x  e.  B  -.  x  e.  A )
 
Theoremdifin2OLD 25772 Represent a set difference as an intersection with a larger difference. (Moved to difin2 3431 in main set.mm and may be deleted by mathbox owner, JM. --NM 31-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  C  ->  ( A  \  B )  =  ( ( C  \  B )  i^i  A ) )
 
TheoremprfunOLD 25773 A function with a domain of two elements. (Moved to funpr 5268 in main set.mm and may be deleted by mathbox owner, JM. --NM 26-Aug-2011.) (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  Fun  { <. A ,  C >. ,  <. B ,  D >. } )
 
Theoremprfv1OLD 25774 The value of a function with a domain of two elements. (Moved to fvpr1 5684 in main set.mm and may be deleted by mathbox owner, JM. --NM 3-Sep-2011.) (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  A )  =  C )
 
Theoremprfv2OLD 25775 The value of a function with a domain of two elements. (Moved to fvpr2 5685 in main set.mm and may be deleted by mathbox owner, JM. --NM 3-Sep-2011.) (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  B )  =  D )
 
TheoremprfOLD 25776 A function with a domain of two elements. (Moved to fpr 5666 in main set.mm and may be deleted by mathbox owner, JM. --NM 3-Sep-2011.) (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  { <. A ,  C >. ,  <. B ,  D >. } : { A ,  B } --> { C ,  D }
 )
 
Theorembrabg2 25777* Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   &    |-  ( ch  ->  A  e.  C )   =>    |-  ( B  e.  D  ->  ( A R B  <->  ch ) )
 
TheoreminpreimaOLD 25778 Preimage of an intersection. (Moved to inpreima 5614 in main set.mm and may be deleted by mathbox owner, JM. --NM 28-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Fun  F  ->  ( `' F " ( A  i^i  B ) )  =  ( ( `' F " A )  i^i  ( `' F " B ) ) )
 
TheoremunpreimaOLD 25779 Preimage of a union. (Moved to unpreima 5613 in main set.mm and may be deleted by mathbox owner, JM. --NM 29-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Fun  F  ->  ( `' F " ( A  u.  B ) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
 
TheoremrespreimaOLD 25780 The preimage of a restricted function. (Moved to respreima 5616 in main set.mm and may be deleted by mathbox owner, JM. --NM 30-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Fun  F  ->  ( `' ( F  |`  B )
 " A )  =  ( ( `' F " A )  i^i  B ) )
 
TheoremfoelrnOLD 25781* Property of a surjective function. (Moved to foelrn 5641 in main set.mm and may be deleted by mathbox owner, JM. --NM 4-Feb-2013.) (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( F : A -onto-> B  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
 
Theoremfoco2OLD 25782 If a composition of two functions is surjective, then the function on the left is surjective. (Moved to foco2 5642 in main set.mm and may be deleted by mathbox owner, JM. --NM 18-Apr-2013.) (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( F : B --> C  /\  G : A --> B  /\  ( F  o.  G ) : A -onto-> C )  ->  F : B -onto-> C )
 
TheoremfnimaprOLD 25783 The image of a pair under a funtion. (Moved to fnimapr 5545 in main set.mm and may be deleted by mathbox owner, JM. --NM 19-Apr-2013.) (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( F " { B ,  C }
 )  =  { ( F `  B ) ,  ( F `  C ) } )
 
Theoremopelopab3 25784* Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  ( ch  ->  A  e.  C )   =>    |-  ( B  e.  D  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch ) )
 
Theoremcocanfo 25785 Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  (
 ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B )  /\  ( G  o.  F )  =  ( H  o.  F ) ) 
 ->  G  =  H )
 
TheoremxpengOLD 25786 Equinumerosity law for cross product. (Moved to xpen 7020 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 17-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  W  /\  B  e.  X )  /\  ( C  e.  Y  /\  D  e.  Z ) )  ->  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A  X.  C ) 
 ~~  ( B  X.  D ) ) )
 
TheoremfvifOLD 25787 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to fvif 5501 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( F `  if ( ph ,  A ,  B ) )  =  if ( ph ,  ( F `  A ) ,  ( F `  B ) )
 
Theoremifeq1daOLD 25788 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to ifeq1da 3591 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
 
Theoremifeq2daOLD 25789 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to ifeq2da 3592 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  -.  ps )  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
 
TheoremifcldaOLD 25790 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to ifclda 3593 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  A  e.  C )   &    |-  ( ( ph  /\  -.  ps )  ->  B  e.  C )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )
 
Theoremxpeq1dOLD 25791 Equality deduction for cross product. (Moved to xpeq1d 4711 in main set.mm and may be deleted by mathbox owner, JM. --NM 4-Feb-2013.) (Contributed by Jeff Madsen, 17-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2dOLD 25792 Equality deduction for cross product. (Moved to xpeq1d 4711 in main set.mm and may be deleted by mathbox owner, JM. --NM 4-Feb-2013.) (Contributed by Jeff Madsen, 17-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
TheoremresexOLD 25793 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.) (Moved to resex 4994 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( A  |`  B )  e.  _V
 
Theorembrresi 25794 Restriction of a binary relation. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  B  e.  _V   =>    |-  ( A ( R  |`  C ) B  ->  A R B )
 
Theoremopabex3OLD 25795* Existence of an ordered pair abstraction. (Moved to opabex3 5731 in main set.mm and may be deleted by mathbox owner, JM. --NM 30-Jan-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  ( x  e.  A  ->  { y  |  ph }  e.  _V )   =>    |- 
 { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
 
Theoremfnopabeqd 25796* Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) } )
 
Theoremfvopabf4g 25797* Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  C  e.  _V   &    |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  ( R 
 ^m  D )  |->  B )   =>    |-  ( ( D  e.  X  /\  R  e.  Y  /\  A : D --> R ) 
 ->  ( F `  A )  =  C )
 
Theoremeqfnun 25798 Two functions on  A  u.  B are equal if and only if they have equal restrictions to both  A and  B. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  (
 ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  ->  ( F  =  G  <->  ( ( F  |`  A )  =  ( G  |`  A ) 
 /\  ( F  |`  B )  =  ( G  |`  B ) ) ) )
 
Theoremfnopabco 25799* Composition of a function with a function abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  ( x  e.  A  ->  B  e.  C )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }   &    |-  G  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( H `  B ) ) }   =>    |-  ( H  Fn  C  ->  G  =  ( H  o.  F ) )
 
Theoremopropabco 25800* Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
 |-  ( x  e.  A  ->  B  e.  R )   &    |-  ( x  e.  A  ->  C  e.  S )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  = 
 <. B ,  C >. ) }   &    |-  G  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( B M C ) ) }   =>    |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
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