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Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrelmpt2opab 6201* Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  { <. z ,  w >.  |  ph } )   =>    |-  Rel  ( C F D )
 
Theoremfmpt2co 6202* Composition of two functions. Variation of fmptco 5691 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  R  e.  C )   &    |-  ( ph  ->  F  =  ( x  e.  A ,  y  e.  B  |->  R ) )   &    |-  ( ph  ->  G  =  ( z  e.  C  |->  S ) )   &    |-  (
 z  =  R  ->  S  =  T )   =>    |-  ( ph  ->  ( G  o.  F )  =  ( x  e.  A ,  y  e.  B  |->  T ) )
 
Theoremoprabco 6203* Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  D )   &    |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( H `  C ) )   =>    |-  ( H  Fn  D  ->  G  =  ( H  o.  F ) )
 
Theoremoprab2co 6204* Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  R )   &    |-  ( ( x  e.  A  /\  y  e.  B )  ->  D  e.  S )   &    |-  F  =  ( x  e.  A ,  y  e.  B  |->  <. C ,  D >. )   &    |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( C M D ) )   =>    |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
 
Theoremdf1st2 6205* An alternate possible definition of the  1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
 
Theoremdf2nd2 6206* An alternate possible definition of the  2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  z  =  y }  =  ( 2nd  |`  ( _V  X.  _V ) )
 
Theorem1stconst 6207 The mapping of a restriction of the  1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
 |-  ( B  e.  V  ->  ( 1st  |`  ( A  X.  { B }
 ) ) : ( A  X.  { B } ) -1-1-onto-> A )
 
Theorem2ndconst 6208 The mapping of a restriction of the  2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
 |-  ( A  e.  V  ->  ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -1-1-onto-> B )
 
Theoremdfmpt2 6209* Alternate definition for the "maps to" notation df-mpt2 5863 (although it requires that  C be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  C  e.  _V   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  = 
 U_ x  e.  A  U_ y  e.  B  { <.
 <. x ,  y >. ,  C >. }
 
Theoremcurry1 6210* Composition with  `' ( 2nd  |`  ( { C }  X.  _V ) ) turns any binary operation  F with a constant first operand into a function  G of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
 
Theoremcurry1val 6211 The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  A )  ->  ( G `  D )  =  ( C F D ) )
 
Theoremcurry1f 6212 Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)
 |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )   =>    |-  ( ( F :
 ( A  X.  B )
 --> D  /\  C  e.  A )  ->  G : B
 --> D )
 
Theoremcurry2 6213* Composition with  `' ( 1st  |`  ( _V  X.  { C }
) ) turns any binary operation  F with a constant second operand into a function  G of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.)
 |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
 
Theoremcurry2f 6214 Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
 |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )   =>    |-  ( ( F :
 ( A  X.  B )
 --> D  /\  C  e.  B )  ->  G : A
 --> D )
 
Theoremcurry2val 6215 The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
 |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  B )  ->  ( G `  D )  =  ( D F C ) )
 
Theoremcnvf1olem 6216 Lemma for cnvf1o 6217. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) ) 
 ->  ( C  e.  `' A  /\  B  =  U. `' { C } )
 )
 
Theoremcnvf1o 6217* Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( Rel  A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
 
Theoremfparlem1 6218 Lemma for fpar 6222. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  =  ( { x }  X.  _V )
 
Theoremfparlem2 6219 Lemma for fpar 6222. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " {
 y } )  =  ( _V  X.  {
 y } )
 
Theoremfparlem3 6220* Lemma for fpar 6222. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( F  Fn  A  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V 
 X.  _V ) ) ) )  =  U_ x  e.  A  ( ( { x }  X.  _V )  X.  ( { ( F `
  x ) }  X.  _V ) ) )
 
Theoremfparlem4 6221* Lemma for fpar 6222. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( G  Fn  B  ->  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V 
 X.  _V ) ) ) )  =  U_ y  e.  B  ( ( _V 
 X.  { y } )  X.  ( _V  X.  {
 ( G `  y
 ) } ) ) )
 
Theoremfpar 6222* Merge two functions in parallel. Use as the second argument of a composition with a (2-place) operation to build compound operations such as  z  =  ( ( sqr `  x
)  +  ( abs `  y ) ). (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  H  =  ( ( `' ( 1st  |`  ( _V 
 X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V 
 X.  _V ) ) ) ) )   =>    |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  H  =  ( x  e.  A ,  y  e.  B  |->  <. ( F `
  x ) ,  ( G `  y
 ) >. ) )
 
Theoremfsplit 6223 A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 6222 in order to build compound functions such as  y  =  ( ( sqr `  x
)  +  ( abs `  x ) ). (Contributed by NM, 17-Sep-2007.)
 |-  `' ( 1st  |`  _I  )  =  ( x  e.  _V  |->  <. x ,  x >. )
 
Theoremalgrflem 6224 Lemma for algrf 12743 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( B ( F  o.  1st ) C )  =  ( F `
  B )
 
Theoremfrxp 6225* A lexicographical ordering of two well founded classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) (Proof shortened by Wolf Lammen, 4-Oct-2014.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
 )  \/  ( ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x ) S ( 2nd `  y )
 ) ) ) }   =>    |-  (
 ( R  Fr  A  /\  S  Fr  B ) 
 ->  T  Fr  ( A  X.  B ) )
 
Theoremxporderlem 6226* Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
 )  \/  ( ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x ) S ( 2nd `  y )
 ) ) ) }   =>    |-  ( <. a ,  b >. T
 <. c ,  d >.  <->  (
 ( ( a  e.  A  /\  c  e.  A )  /\  (
 b  e.  B  /\  d  e.  B )
 )  /\  ( a R c  \/  (
 a  =  c  /\  b S d ) ) ) )
 
Theorempoxp 6227* A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
 )  \/  ( ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x ) S ( 2nd `  y )
 ) ) ) }   =>    |-  (
 ( R  Po  A  /\  S  Po  B ) 
 ->  T  Po  ( A  X.  B ) )
 
Theoremsoxp 6228* A lexicographical ordering of two strictly ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
 )  \/  ( ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x ) S ( 2nd `  y )
 ) ) ) }   =>    |-  (
 ( R  Or  A  /\  S  Or  B ) 
 ->  T  Or  ( A  X.  B ) )
 
Theoremwexp 6229* A lexicographical ordering of two well ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
 )  \/  ( ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x ) S ( 2nd `  y )
 ) ) ) }   =>    |-  (
 ( R  We  A  /\  S  We  B ) 
 ->  T  We  ( A  X.  B ) )
 
Theoremfnwelem 6230* Lemma for fnwe 6231. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  A )  /\  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x S y ) ) ) }   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  R  We  B )   &    |-  ( ph  ->  S  We  A )   &    |-  ( ph  ->  ( F " w )  e.  _V )   &    |-  Q  =  { <. u ,  v >.  |  ( ( u  e.  ( B  X.  A )  /\  v  e.  ( B  X.  A ) )  /\  ( ( 1st `  u ) R ( 1st `  v
 )  \/  ( ( 1st `  u )  =  ( 1st `  v
 )  /\  ( 2nd `  u ) S ( 2nd `  v )
 ) ) ) }   &    |-  G  =  ( z  e.  A  |->  <.
 ( F `  z
 ) ,  z >. )   =>    |-  ( ph  ->  T  We  A )
 
Theoremfnwe 6231* A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  A )  /\  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x S y ) ) ) }   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  R  We  B )   &    |-  ( ph  ->  S  We  A )   &    |-  ( ph  ->  ( F " w )  e.  _V )   =>    |-  ( ph  ->  T  We  A )
 
Theoremfnse 6232* Condition for the well-order in fnwe 6231 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  A )  /\  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x S y ) ) ) }   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  R Se  B )   &    |-  ( ph  ->  ( `' F " w )  e. 
 _V )   =>    |-  ( ph  ->  T Se  A )
 
2.4.14  Function transposition
 
Syntaxctpos 6233 The transposition of a function.
 class tpos  F
 
Definitiondf-tpos 6234* Define the transposition of a function, which is a function  G  = tpos  F satisfying  G ( x ,  y )  =  F ( y ,  x ). (Contributed by Mario Carneiro, 10-Sep-2015.)
 |- tpos  F  =  ( F  o.  ( x  e.  ( `' dom  F  u.  { (/)
 } )  |->  U. `' { x } ) )
 
Theoremtposss 6235 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  C_  G  -> tpos 
 F  C_ tpos  G )
 
Theoremtposeq 6236 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  =  G  -> tpos 
 F  = tpos  G )
 
Theoremtposeqd 6237 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  -> tpos  F  = tpos  G )
 
Theoremtposssxp 6238 The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |- tpos  F  C_  ( ( `'
 dom  F  u.  { (/) } )  X.  ran  F )
 
Theoremreltpos 6239 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |- 
 Rel tpos  F
 
Theorembrtpos2 6240 Value of the transposition at a pair  <. A ,  B >.. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( B  e.  V  ->  ( Atpos  F B  <->  ( A  e.  ( `'
 dom  F  u.  { (/) } )  /\  U. `' { A } F B ) ) )
 
Theorembrtpos0 6241 The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows us to eliminate sethood hypotheses on  A ,  B in brtpos 6243. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  (/) F A ) )
 
Theoremreldmtpos 6242 Necessary and sufficient condition for  dom tpos  F to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom tpos  F  <->  -.  (/)  e.  dom  F )
 
Theorembrtpos 6243 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( C  e.  V  ->  ( <. A ,  B >.tpos  F C  <->  <. B ,  A >. F C ) )
 
Theoremottpos 6244 The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  ( C  e.  V  ->  ( <. A ,  B ,  C >.  e. tpos  F  <->  <. B ,  A ,  C >.  e.  F ) )
 
Theoremrelbrtpos 6245 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015.)
 |-  ( Rel  F  ->  (
 <. A ,  B >.tpos  F C  <->  <. B ,  A >. F C ) )
 
Theoremdmtpos 6246 The domain of tpos  F when  dom  F is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  ->  dom tpos  F  =  `' dom  F )
 
Theoremrntpos 6247 The range of tpos  F when  dom  F is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  ->  ran tpos  F  =  ran  F )
 
Theoremtposexg 6248 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  e.  V  -> tpos 
 F  e.  _V )
 
Theoremovtpos 6249 The transposition swaps the arguments in a two-argument function. When  F is a matrix, which is to say a function from  ( 1 ... m )  X.  (
1 ... n ) to  RR or some ring, tpos  F is the transposition of  F, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Atpos  F B )  =  ( B F A )
 
Theoremtposfun 6250 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Fun  F  ->  Fun tpos  F )
 
Theoremdftpos2 6251* Alternate definition of tpos when 
F has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  -> tpos 
 F  =  ( F  o.  ( x  e.  `' dom  F  |->  U. `' { x } ) ) )
 
Theoremdftpos3 6252* Alternate definition of tpos when 
F has relational domain. Compare df-cnv 4697. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  -> tpos 
 F  =  { <. <. x ,  y >. ,  z >.  |  <. y ,  x >. F z }
 )
 
Theoremdftpos4 6253* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos  F  =  ( F  o.  ( x  e.  (
 ( _V  X.  _V )  u.  { (/) } )  |-> 
 U. `' { x } ) )
 
Theoremtpostpos 6254 Value of the double transposition for a general class  F. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |- tpos tpos  F  =  ( F  i^i  ( ( ( _V 
 X.  _V )  u.  { (/)
 } )  X.  _V ) )
 
Theoremtpostpos2 6255 Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( ( Rel  F  /\  Rel  dom  F )  -> tpos tpos  F  =  F )
 
Theoremtposfn2 6256 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
 
Theoremtposfo2 6257 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A -onto-> B  -> tpos 
 F : `' A -onto-> B ) )
 
Theoremtposf2 6258 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )
 
Theoremtposf12 6259 Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A -1-1-> B  -> tpos 
 F : `' A -1-1-> B ) )
 
Theoremtposf1o2 6260 Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A -1-1-onto-> B  -> tpos  F : `' A
 -1-1-onto-> B ) )
 
Theoremtposfo 6261 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( F : ( A  X.  B )
 -onto-> C  -> tpos  F : ( B  X.  A )
 -onto-> C )
 
Theoremtposf 6262 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( F : ( A  X.  B ) --> C  -> tpos  F : ( B  X.  A ) --> C )
 
Theoremtposfn 6263 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( F  Fn  ( A  X.  B )  -> tpos  F  Fn  ( B  X.  A ) )
 
Theoremtpos0 6264 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
 |- tpos  (/) 
 =  (/)
 
Theoremtposco 6265 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos 
 ( F  o.  G )  =  ( F  o. tpos  G )
 
Theoremtpossym 6266* Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( F  Fn  ( A  X.  A )  ->  (tpos  F  =  F  <->  A. x  e.  A  A. y  e.  A  ( x F y )  =  ( y F x ) ) )
 
Theoremtposeqi 6267 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  G   =>    |- tpos  F  = tpos  G
 
Theoremtposex 6268 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  e.  _V   =>    |- tpos  F  e.  _V
 
Theoremnftpos 6269 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F/_ x F   =>    |-  F/_ xtpos  F
 
Theoremtposoprab 6270* Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  { <. <. x ,  y >. ,  z >.  |  ph }   =>    |- tpos  F  =  { <.
 <. y ,  x >. ,  z >.  |  ph }
 
Theoremtposmpt2 6271* Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |- tpos  F  =  (
 y  e.  B ,  x  e.  A  |->  C )
 
2.4.15  Curry and uncurry
 
Syntaxccur 6272 Extend class notation to include the currying function.
 class curry  A
 
Syntaxcunc 6273 Extend class notation to include the uncurrying function.
 class uncurry  A
 
Definitiondf-cur 6274* Define the currying of  F, which splits a function of two arguments into a function of the first argument, producing a function over the second argument. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- curry  F  =  ( x  e.  dom  dom  F  |->  { <. y ,  z >.  |  <. x ,  y >. F z } )
 
Definitiondf-unc 6275* Define the uncurrying of  F, which takes a function producing functions, and transforms it into a two-argument function. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- uncurry  F  =  { <. <. x ,  y >. ,  z >.  |  y ( F `  x ) z }
 
2.4.16  Proper subset relation
 
Syntaxcrpss 6276 Extend class notation to include the reified proper subset relation.
 class [ C.]
 
Definitiondf-rpss 6277* Define a relation which corresponds to proper subsethood df-pss 3168 on sets. This allows us to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 6282. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- [ C.]  =  { <. x ,  y >.  |  x  C.  y }
 
Theoremrelrpss 6278 The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- 
 Rel [ C.]
 
Theorembrrpssg 6279 The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  ( B  e.  V  ->  ( A [ C.]  B  <->  A 
 C.  B ) )
 
Theorembrrpss 6280 The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  B  e.  _V   =>    |-  ( A [ C.]  B  <->  A  C.  B )
 
Theoremporpss 6281 Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- [ C.]  Po  A
 
Theoremsorpss 6282* Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  ( [ C.]  Or  A  <->  A. x  e.  A  A. y  e.  A  ( x  C_  y  \/  y  C_  x ) )
 
Theoremsorpssi 6283 Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A ) )  ->  ( B 
 C_  C  \/  C  C_  B ) )
 
Theoremsorpssun 6284 A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.)
 |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A ) )  ->  ( B  u.  C )  e.  A )
 
Theoremsorpssin 6285 A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.)
 |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A ) )  ->  ( B  i^i  C )  e.  A )
 
Theoremsorpssuni 6286* In a chain of sets, a maximal element is the union of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  ( [ C.]  Or  Y  ->  ( E. u  e.  Y  A. v  e.  Y  -.  u  C.  v 
 <-> 
 U. Y  e.  Y ) )
 
Theoremsorpssint 6287* In a chain of sets, a minimal element is the intersection of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  ( [ C.]  Or  Y  ->  ( E. u  e.  Y  A. v  e.  Y  -.  v  C.  u 
 <-> 
 |^| Y  e.  Y ) )
 
Theoremsorpsscmpl 6288* The componentwise complement of a chain of sets is also a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  ( [ C.]  Or  Y  -> [
 C.]  Or  { u  e.  ~P A  |  ( A  \  u )  e.  Y } )
 
2.4.17  Iota properties
 
Theoremfvopab5 6289* The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  F  =  { <. x ,  y >.  |  ph }   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( F `  A )  =  ( iota y ps ) )
 
Theoremopiota 6290* The property of a uniquely specified ordered pair. (Contributed by Mario Carneiro, 21-May-2015.)
 |-  I  =  ( iota
 z E. x  e.  A  E. y  e.  B  ( z  = 
 <. x ,  y >.  /\  ph ) )   &    |-  X  =  ( 1st `  I )   &    |-  Y  =  ( 2nd `  I
 )   &    |-  ( x  =  C  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  D  ->  ( ps  <->  ch ) )   =>    |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  =  <. x ,  y >.  /\  ph )  ->  ( ( C  e.  A  /\  D  e.  B  /\  ch )  <->  ( C  =  X  /\  D  =  Y ) ) )
 
Theoremopabiotafun 6291* Define a function whose value is "the unique  y such that  ph ( x ,  y )". (Contributed by NM, 19-May-2015.)
 |-  F  =  { <. x ,  y >.  |  {
 y  |  ph }  =  { y } }   =>    |-  Fun  F
 
Theoremopabiotadm 6292* Define a function whose value is "the unique  y such that  ph ( x ,  y )". (Contributed by NM, 16-Nov-2013.)
 |-  F  =  { <. x ,  y >.  |  {
 y  |  ph }  =  { y } }   =>    |-  dom  F  =  { x  |  E! y ph }
 
Theoremopabiota 6293* Define a function whose value is "the unique  y such that  ph ( x ,  y )". (Contributed by NM, 16-Nov-2013.)
 |-  F  =  { <. x ,  y >.  |  {
 y  |  ph }  =  { y } }   &    |-  ( x  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( B  e.  dom 
 F  ->  ( F `  B )  =  (
 iota y ps )
 )
 
2.4.18  Cantor's Theorem
 
Theoremcanth 6294 No set  A is equinumerous to its power set (Cantor's theorem), i.e. no function can map  A it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 7014. Note that  A must be a set: this theorem does not hold when  A is too large to be a set; see ncanth 6295 for a counterexample. (Use nex 1542 if you want the form  -.  E. f f : A -onto-> ~P A.) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
 |-  A  e.  _V   =>    |-  -.  F : A -onto-> ~P A
 
Theoremncanth 6295 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4152). Specifically, the identity function maps the universe onto its power class. Compare canth 6294 that works for sets. See also the remark in ru 2990 about NF, in which Cantor's theorem fails for sets that are "too large." This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004.)
 |- 
 _I  : _V -onto-> ~P _V
 
2.4.19  Undefined values and restricted iota (description binder)
 
Syntaxcund 6296 Extend class notation with undefined value function.
 class  Undef
 
Syntaxcrio 6297 Extend class notation with restricted description binder.
 class  ( iota_ x  e.  A ph )
 
Definitiondf-undef 6298 Define the undefined value function, whose value at set  s is guaranteed not to be a member of 
s (see pwuninel 6300). (Contributed by NM, 15-Sep-2011.)
 |- 
 Undef  =  ( s  e.  _V  |->  ~P U. s )
 
Theorempwuninel2 6299 Direct proof of pwuninel 6300 avoiding functions and thus several ZF axioms. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( U. A  e.  V  ->  -.  ~P U. A  e.  A )
 
Theorempwuninel 6300 The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. See also pwuninel2 6299. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |- 
 -.  ~P U. A  e.  A
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