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Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembrtpos2 6201 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 6202 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 6204. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  (/) F A ) )
 
Theoremreldmtpos 6203 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 6204 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 6205 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 6206 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 6207 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 6208 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 6209 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  e.  V  -> tpos 
 F  e.  _V )
 
Theoremovtpos 6210 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 6211 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Fun  F  ->  Fun tpos  F )
 
Theoremdftpos2 6212* 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 6213* Alternate definition of tpos when 
F has relational domain. Compare df-cnv 4696. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  -> tpos 
 F  =  { <. <. x ,  y >. ,  z >.  |  <. y ,  x >. F z }
 )
 
Theoremdftpos4 6214* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos  F  =  ( F  o.  ( x  e.  (
 ( _V  X.  _V )  u.  { (/) } )  |-> 
 U. `' { x } ) )
 
Theoremtpostpos 6215 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 6216 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 6217 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
 
Theoremtposfo2 6218 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A -onto-> B  -> tpos 
 F : `' A -onto-> B ) )
 
Theoremtposf2 6219 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )
 
Theoremtposf12 6220 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 6221 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 6222 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 6223 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 6224 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( F  Fn  ( A  X.  B )  -> tpos  F  Fn  ( B  X.  A ) )
 
Theoremtpos0 6225 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
 |- tpos  (/) 
 =  (/)
 
Theoremtposco 6226 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos 
 ( F  o.  G )  =  ( F  o. tpos  G )
 
Theoremtpossym 6227* 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 6228 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  G   =>    |- tpos  F  = tpos  G
 
Theoremtposex 6229 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  e.  _V   =>    |- tpos  F  e.  _V
 
Theoremnftpos 6230 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F/_ x F   =>    |-  F/_ xtpos  F
 
Theoremtposoprab 6231* 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 6232* 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 )
 
2.4.14  Curry and uncurry
 
Syntaxccur 6233 Extend class notation to include the currying function.
 class curry  A
 
Syntaxcunc 6234 Extend class notation to include the uncurrying function.
 class uncurry  A
 
Definitiondf-cur 6235* 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 6236* 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 }
 
2.4.15  Proper subset relation
 
Syntaxcrpss 6237 Extend class notation to include the reified proper subset relation.
 class [ C.]
 
Definitiondf-rpss 6238* Define a relation which corresponds to proper subsethood df-pss 3169 on sets. This allows us to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 6243. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- [ C.]  =  { <. x ,  y >.  |  x  C.  y }
 
Theoremrelrpss 6239 The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- 
 Rel [ C.]
 
Theorembrrpssg 6240 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 6241 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 6242 Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- [ C.]  Po  A
 
Theoremsorpss 6243* 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 6244 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 6245 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 6246 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 6247* 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 6248* 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 6249* 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 } )
 
2.4.16  Definite description binder (inverted iota)
 
Syntaxcio 6250 Extend class notation with Russell's definition description binder (inverted iota).
 class  ( iota x ph )
 
Theoremiotajust 6251* Soundness justification theorem for df-iota 6252. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |- 
 U. { y  |  { x  |  ph }  =  { y } }  =  U. { z  |  { x  |  ph }  =  { z } }
 
Definitiondf-iota 6252* Define Russell's definition description binder, which can be read as "the unique  x such that  ph," where  ph ordinarily contains  x as a free variable. Our definition is meaningful only when there is exactly one  x such that  ph is true (see iotaval 6263); otherwise, it evaluates to the empty set (see iotanul 6267). Russell used the inverted iota symbol 
iota to represent the binder.

Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use riotacl2 6313 (or iotacl 6275 for unbounded iota), as demonstrated in the proof of supub 7205. This can be easier than applying riotasbc 6315 or a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF.

(Contributed by Andrew Salmon, 30-Jun-2011.)

 |-  ( iota x ph )  =  U. { y  |  { x  |  ph }  =  { y } }
 
Theoremdfiota2 6253* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( iota x ph )  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
 
Theoremnfiota1 6254 Bound-variable hypothesis builder for the  iota class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x ( iota x ph )
 
Theoremnfiotad 6255 Deduction version of nfiota 6256. (Contributed by NM, 18-Feb-2013.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/_ x ( iota y ps ) )
 
Theoremnfiota 6256 Bound-variable hypothesis builder for the  iota class. (Contributed by NM, 23-Aug-2011.)
 |- 
 F/ x ph   =>    |-  F/_ x ( iota y ph )
 
Theoremcbviota 6257 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  F/ y ph   &    |-  F/ x ps   =>    |-  ( iota x ph )  =  ( iota y ps )
 
Theoremcbviotav 6258* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( iota x ph )  =  ( iota
 y ps )
 
Theoremsb8iota 6259 Variable substitution in description binder. Compare sb8eu 2162. (Contributed by NM, 18-Mar-2013.)
 |- 
 F/ y ph   =>    |-  ( iota x ph )  =  ( iota y [ y  /  x ] ph )
 
Theoremiotaeq 6260 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )
 
Theoremiotabi 6261 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )
 
Theoremuniabio 6262* Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x (
 ph 
 <->  x  =  y ) 
 ->  U. { x  |  ph
 }  =  y )
 
Theoremiotaval 6263* Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x (
 ph 
 <->  x  =  y ) 
 ->  ( iota x ph )  =  y )
 
Theoremiotauni 6264 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  =  U. { x  |  ph } )
 
Theoremiotaint 6265 Equivalence between two different forms of  iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E! x ph  ->  ( iota x ph )  =  |^| { x  |  ph } )
 
Theoremiota1 6266 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
 |-  ( E! x ph  ->  ( ph  <->  ( iota x ph )  =  x ) )
 
Theoremiotanul 6267 Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one  x that satisfies  ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( -.  E! x ph 
 ->  ( iota x ph )  =  (/) )
 
Theoremiotassuni 6268 The  iota class is a subset of the union of all elements satisfying  ph. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( iota x ph )  C_  U. { x  |  ph }
 
Theoremiotaex 6269 Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the  iota class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( iota x ph )  e.  _V
 
Theoremiota4 6270 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  -> 
 [. ( iota x ph )  /  x ]. ph )
 
Theoremiota4an 6271 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x (
 ph  /\  ps )  -> 
 [. ( iota x ( ph  /\  ps )
 )  /  x ]. ph )
 
Theoremiota5 6272* A method for computing iota. (Contributed by NM, 17-Sep-2013.)
 |-  ( ( ph  /\  A  e.  V )  ->  ( ps 
 <->  x  =  A ) )   =>    |-  ( ( ph  /\  A  e.  V )  ->  ( iota x ps )  =  A )
 
Theoremiotabidv 6273* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota x ps )  =  ( iota x ch ) )
 
Theoremiotabii 6274 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( ph  <->  ps )   =>    |-  ( iota x ph )  =  ( iota x ps )
 
Theoremiotacl 6275 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 6252). If you have a bounded iota-based definition, riotacl2 6313 may be useful.

(Contributed by Andrew Salmon, 1-Aug-2011.)

 |-  ( E! x ph  ->  ( iota x ph )  e.  { x  |  ph } )
 
Theoremiota2df 6276 A condition that allows us to represent "the unique element such that  ph " with a class expression  A. (Contributed by NM, 30-Dec-2014.)
 |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  E! x ps )   &    |-  (
 ( ph  /\  x  =  B )  ->  ( ps 
 <->  ch ) )   &    |-  F/ x ph   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
 
Theoremiota2d 6277* A condition that allows us to represent "the unique element such that  ph " with a class expression  A. (Contributed by NM, 30-Dec-2014.)
 |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  E! x ps )   &    |-  (
 ( ph  /\  x  =  B )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
 
Theoremiota2 6278* The unique element such that 
ph. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  E! x ph )  ->  ( ps 
 <->  ( iota x ph )  =  A )
 )
 
Theoremsniota 6279 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  ( E! x ph  ->  { x  |  ph }  =  { ( iota
 x ph ) } )
 
Theoremdffv3 6280* A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( F `  A )  =  ( iota x x  e.  ( F
 " { A }
 ) )
 
Theoremfv4 6281* Alternate definition of the value of a function. The value of a function expressed using 
iota. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( F `  A )  =  ( iota x A F x )
 
Theoremfvopab5 6282* 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 6283* 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 6284* 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 6285* 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 6286* 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 )
 )
 
2.4.17  Cantor's Theorem
 
Theoremcanth 6287 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 7009. Note that  A must be a set: this theorem does not hold when  A is too large to be a set; see ncanth 6288 for a counterexample. (Use nex 1547 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 6288 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4153). Specifically, the identity function maps the universe onto its power class. Compare canth 6287 that works for sets. See also the remark in ru 2991 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
 
2.4.18  Undefined values and restricted iota (description binder)
 
Syntaxcund 6289 Extend class notation with undefined value function.
 class  Undef
 
Syntaxcrio 6290 Extend class notation with restricted description binder.
 class  ( iota_ x  e.  A ph )
 
Definitiondf-undef 6291 Define the undefined value function, whose value at set  s is guaranteed not to be a member of 
s (see pwuninel 6295). (Contributed by NM, 15-Sep-2011.)
 |- 
 Undef  =  ( s  e.  _V  |->  ~P U. s )
 
Theorempwnss 6292 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( A  e.  V  ->  -.  ~P A  C_  A )
 
Theorempwne 6293 No set equals its power set. The sethood antecedent is necessary; compare pwv 3827. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  ( A  e.  V  ->  ~P A  =/=  A )
 
TheorempwuninelALT 6294 Direct proof of pwuninel 6295 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 6295 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. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |- 
 -.  ~P U. A  e.  A
 
Theoremundefval 6296 Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 6298 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
 
Theoremundefnel2 6297 The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.)
 |-  ( S  e.  V  ->  -.  ( Undef `  S )  e.  S )
 
Theoremundefnel 6298 The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.)
 |-  ( S  e.  V  ->  ( Undef `  S )  e/  S )
 
Definitiondf-riota 6299 Define restricted description binder. In case it doesn't exist, we return a set which is not a member of the domain of discourse  A. See also comments for df-iota 6252. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( iota_ x  e.  A ph )  =  if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  ( Undef `  { x  |  x  e.  A } ) )
 
Theoremriotaeqdv 6300* Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( iota_ x  e.  A ps )  =  ( iota_ x  e.  B ps ) )
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