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Theorem List for Metamath Proof Explorer - 26401-26500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremptfinfin 26401* 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 26402 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 26403 A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  ( A  e.  ( LocFin `  J )  ->  J  e.  Top )
 
Theoremlocfinbas 26404 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 26405* 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 26406 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 26407 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 26408 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 26409 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 26410* 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 26411* 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 26412* Lemma for neibastop2 26413. (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 26413* 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 26414* 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 26415 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 26416* 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 26417* 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 26418* 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 26419* 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 26420* 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 26421* 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 26422 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 26423* The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 17553. (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 26424* 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 26425 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 26426 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 26427 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 26428 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 26429 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 26430* Lemma for filnet 26434. 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 26431* Lemma for filnet 26434. 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 26432* Lemma for filnet 26434. (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 26433* Lemma for filnet 26434. (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 26434* 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 26435 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 26436 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 26437 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 26438 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 26439 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 26440 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 26441* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Moved to ralrimivva 2648 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 26442* Substitution into "at most one". (Moved to sbmo 2186 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 26443* Distribute quantification over "or". (Moved to 2ralor 2722 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 26444 Restricted existential quantification over union. (Moved to rexun 3368 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 26445* Restricted quantification over union. (Moved to ralun 3370 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 26446* Restricted existential quantification over a singleton. (Moved to rexsn 3688 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 26447* Specialized existential commutation lemma. (Moved to rexcom4a 2821 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 26448* Specialized existential commutation lemma. (Moved to rexcom4b 2822 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 26449* Rearrange three existential quantifiers. (Moved to 3reeanv 2721 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 26450* Derive membership from uniqueness. (Moved to morex 2962 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 26451* The unique element such that 
ph. (Moved to iota2 5261 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 26452* 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 26453* Equality of restricted class abstractions. (Moved to rabeqbidv 2796 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 26454* Condition for a restricted class abstraction to be empty. (Moved to rabeq0 3489 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 26455* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) (Moved to rabxm 3490 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 26456* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) (Moved to rabnc 3491 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 26457* Universal quantification over a class abstraction. (Moved to ralab 2939 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 26458* Universal quantification over a restricted class abstraction. (Moved to ralrab 2940 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 26459* Existential quantification over a class abstraction. (Moved to rexrab 2942 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 26460* Change the domain of quantification by a function. (Moved to ralima 5774 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 26461* 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 26462* 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 26463* Two ways of saying that two classes are disjoint. (Moved to disjr 3509 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 26464 Represent a set difference as an intersection with a larger difference. (Moved to difin2 3443 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 26465 A function with a domain of two elements. (Moved to funpr 5318 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 26466 The value of a function with a domain of two elements. (Moved to fvpr1 5738 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 26467 The value of a function with a domain of two elements. (Moved to fvpr2 5739 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 26468 A function with a domain of two elements. (Moved to fpr 5720 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 26469* 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 26470 Preimage of an intersection. (Moved to inpreima 5668 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 26471 Preimage of a union. (Moved to unpreima 5667 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 26472 The preimage of a restricted function. (Moved to respreima 5670 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 26473* Property of a surjective function. (Moved to foelrn 5695 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 26474 If a composition of two functions is surjective, then the function on the left is surjective. (Moved to foco2 5696 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 26475 The image of a pair under a funtion. (Moved to fnimapr 5599 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 26476* 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 26477 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 26478 Equinumerosity law for cross product. (Moved to xpen 7040 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 26479 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to fvif 5556 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 26480 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to ifeq1da 3603 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 26481 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to ifeq2da 3604 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 26482 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to ifclda 3605 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 26483 Equality deduction for cross product. (Moved to xpeq1d 4728 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 26484 Equality deduction for cross product. (Moved to xpeq1d 4728 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 26485 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.) (Moved to resex 5011 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 26486 Restriction of a binary relation. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  B  e.  _V   =>    |-  ( A ( R  |`  C ) B  ->  A R B )
 
Theoremopabex3OLD 26487* Existence of an ordered pair abstraction. (Moved to opabex3 5785 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 26488* 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 26489* 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 26490 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 26491* 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 26492* 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 ) )
 
TheoremoprabexdOLD 26493* Existence of an operator abstraction. (Moved to oprabexd 5976 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  E* z ps )   &    |-  ( ph  ->  F  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) } )   =>    |-  ( ph  ->  F  e.  _V )
 
Theoremf1opr 26494* Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( F : ( A  X.  B ) -1-1-> C  <->  ( F :
 ( A  X.  B )
 --> C  /\  A. r  e.  A  A. s  e.  B  A. t  e.  A  A. u  e.  B  ( ( r F s )  =  ( t F u )  ->  ( r  =  t  /\  s  =  u ) ) ) )
 
TheoremcnvcanOLD 26495 Composition with the converse. (Moved to funcocnv2 5514 in main set.mm and may be deleted by mathbox owner, JM. --NM 28-Feb-2015.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Fun  F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
 
Theoremcocnv 26496 Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( Fun  F  /\  Fun 
 G )  ->  (
 ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
 
Theoremf1ocan1fv 26497 Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  (
 ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  ( ( F  o.  G ) `
  ( `' G `  X ) )  =  ( F `  X ) )
 
Theoremf1ocan2fv 26498 Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  ( ( F  o.  `' G ) `  ( G `  X ) )  =  ( F `  X ) )
 
Theoremf1elimaOLD 26499 Membership in the image of a 1-1 map. (Moved to f1elima 5803 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( F : A -1-1-> B 
 /\  X  e.  A  /\  Y  C_  A )  ->  ( ( F `  X )  e.  ( F " Y )  <->  X  e.  Y ) )
 
Theoremenf1f1oOLD 26500 A one-to-one mapping of finite sets with the same cardinality is bijective. (Moved to f1finf1o 7102 in main set.mm and may be deleted by mathbox owner, JM. --NM 24-Sep-2013.) (Contributed by Jeff Madsen, 5-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  Fin  /\  B  ~~  A ) 
 ->  ( F : A -1-1-> B 
 ->  F : A -1-1-onto-> B ) )
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