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Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremporpss 6201 Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- [ C.]  Po  A
 
Theoremsorpss 6202* 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 6203 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 6204 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 6205 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 6206* 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 6207* 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 6208* 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 } )
 
2.4.15  Definite description binder (inverted iota)
 
Syntaxcio 6209 Extend class notation with Russell's definition description binder (inverted iota).
 class  ( iota x ph )
 
Theoremiotajust 6210* Soundness justification theorem for df-iota 6211. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |- 
 U. { y  |  { x  |  ph }  =  { y } }  =  U. { z  |  { x  |  ph }  =  { z } }
 
Definitiondf-iota 6211* 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 6222); otherwise, it evaluates to the empty set (see iotanul 6226). Russell used the inverted iota symbol 
iota to represent the binder. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( iota x ph )  =  U. { y  |  { x  |  ph }  =  { y } }
 
Theoremdfiota2 6212* 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 6213 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 6214 Deduction version of nfiota 6215. (Contributed by NM, 18-Feb-2013.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/_ x ( iota y ps ) )
 
Theoremnfiota 6215 Bound-variable hypothesis builder for the  iota class. (Contributed by NM, 23-Aug-2011.)
 |- 
 F/ x ph   =>    |-  F/_ x ( iota y ph )
 
Theoremcbviota 6216 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 6217* 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 6218 Variable substitution in description binder. Compare sb8eu 2135. (Contributed by NM, 18-Mar-2013.)
 |- 
 F/ y ph   =>    |-  ( iota x ph )  =  ( iota y [ y  /  x ] ph )
 
Theoremiotaeq 6219 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )
 
Theoremiotabi 6220 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )
 
Theoremuniabio 6221* 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 6222* 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 6223 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  =  U. { x  |  ph } )
 
Theoremiotaint 6224 Equivalence between two different forms of  iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E! x ph  ->  ( iota x ph )  =  |^| { x  |  ph } )
 
Theoremiota1 6225 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 6226 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 6227 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 6228 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 6229 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  -> 
 [. ( iota x ph )  /  x ]. ph )
 
Theoremiota4an 6230 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 6231* 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 6232* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota x ps )  =  ( iota x ch ) )
 
Theoremiotabii 6233 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( ph  <->  ps )   =>    |-  ( iota x ph )  =  ( iota x ps )
 
Theoremiotacl 6234 Membership law for descriptions. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  e.  { x  |  ph } )
 
Theoremiota2df 6235 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 6236* 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 6237* 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 6238 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 6239* 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 6240* 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 6241* 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 6242* 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 6243* 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 6244* 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 6245* 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 )
 )
 
2.4.16  Cantor's Theorem
 
Theoremcanth 6246 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 6968. Note that  A must be a set: this theorem does not hold when  A is too large to be a set; see ncanth 6247 for a counterexample. (Use nex 1587 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 6247 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4112). Specifically, the identity function maps the universe onto its power class. Compare canth 6246 that works for sets. See also the remark in ru 2951 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
 
2.4.17  Undefined values and restricted iota (description binder)
 
Syntaxcund 6248 Extend class notation with undefined value function.
 class  Undef
 
Syntaxcrio 6249 Extend class notation with restricted description binder.
 class  ( iota_ x  e.  A ph )
 
Definitiondf-undef 6250 Define the undefined value function, whose value at set  s is guaranteed not to be a member of 
s (see pwuninel 6254). (Contributed by NM, 15-Sep-2011.)
 |- 
 Undef  =  ( s  e.  _V  |->  ~P U. s )
 
Theorempwnss 6251 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 6252 No set equals its power set. The sethood antecedent is necessary; compare pwv 3786. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  ( A  e.  V  ->  ~P A  =/=  A )
 
TheorempwuninelALT 6253 Direct proof of pwuninel 6254 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 6254 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 6255 Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 6257 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
 
Theoremundefnel2 6256 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 6257 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 6258 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 6211. (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 6259* 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 ) )
 
Theoremriotabidv 6260* Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota_ x  e.  A ps )  =  ( iota_ x  e.  A ch ) )
 
Theoremriotaeqbidv 6261* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 iota_ x  e.  A ps )  =  ( iota_ x  e.  B ch ) )
 
Theoremriotaex 6262 Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)
 |-  ( iota_ x  e.  A ps )  e.  _V
 
Theoremriotav 6263 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
 |-  ( iota_ x  e.  _V ph )  =  ( iota
 x ph )
 
Theoremriotaiota 6264 Restricted iota in terms of iota. (Contributed by NM, 15-Sep-2011.)
 |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph ) ) )
 
Theoremriotauni 6265 Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
 |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  U. { x  e.  A  |  ph } )
 
Theoremnfriota1 6266* The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x ( iota_ x  e.  A ph )
 
Theoremnfriotad 6267 Deduction version of nfriota 6268. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x (
 iota_ y  e.  A ps ) )
 
Theoremnfriota 6268* A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
 |- 
 F/ x ph   &    |-  F/_ x A   =>    |-  F/_ x ( iota_ y  e.  A ph )
 
Theoremcbvriota 6269* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( iota_ x  e.  A ph )  =  ( iota_ y  e.  A ps )
 
Theoremcbvriotav 6270* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( iota_ x  e.  A ph )  =  ( iota_ y  e.  A ps )
 
Theoremcsbriotag 6271* Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph )
 )
 
Theoremriotacl2 6272 Membership law for "the unique element in  A such that  ph." (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
 |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  { x  e.  A  |  ph } )
 
Theoremriotacl 6273* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
 |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  A )
 
Theoremriotasbc 6274 Substitution law for descriptions. Compare iotasbc 26973. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  ( E! x  e.  A  ph  ->  [. ( iota_ x  e.  A ph )  /  x ]. ph )
 
Theoremriotabidva 6275* Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2748 analog.) (Contributed by NM, 17-Jan-2012.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota_ x  e.  A ps )  =  ( iota_ x  e.  A ch ) )
 
Theoremriotabiia 6276 Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2747 analog.) (Contributed by NM, 16-Jan-2012.)
 |-  ( x  e.  A  ->  ( ph  <->  ps ) )   =>    |-  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  A ps )
 
Theoremriota1 6277* Property of restricted iota. Compare iota1 6225. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota_ x  e.  A ph )  =  x ) )
 
Theoremriota1a 6278 Property of iota. (Contributed by NM, 23-Aug-2011.)
 |-  ( ( x  e.  A  /\  E! x  e.  A  ph )  ->  ( ph  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
 
Theoremriota2df 6279* A deduction version of riota2f 6280. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/_ x B )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ( ph  /\  x  =  B ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ( ph  /\ 
 E! x  e.  A  ps )  ->  ( ch  <->  (
 iota_ x  e.  A ps )  =  B ) )
 
Theoremriota2f 6280* This theorem shows a condition that allows us to represent a descriptor with a class expression  B. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x B   &    |-  F/ x ps   &    |-  ( x  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  ( iota_ x  e.  A ph )  =  B ) )
 
Theoremriota2 6281* This theorem shows a condition that allows us to represent a descriptor with a class expression  B. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  ( iota_ x  e.  A ph )  =  B ) )
 
Theoremriotaprop 6282* Properties of a restricted definite description operator. Todo: can some uses of riota2f 6280 be shortened with this? (Contributed by NM, 23-Nov-2013.)
 |- 
 F/ x ps   &    |-  B  =  ( iota_ x  e.  A ph )   &    |-  ( x  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  ->  ( B  e.  A  /\  ps ) )
 
Theoremriota5f 6283* A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  F/_ x B )   &    |-  ( ph  ->  B  e.  A )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( ps 
 <->  x  =  B ) )   =>    |-  ( ph  ->  ( iota_ x  e.  A ps )  =  B )
 
Theoremriota5 6284* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
 |-  ( ph  ->  B  e.  A )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  x  =  B ) )   =>    |-  ( ph  ->  ( iota_ x  e.  A ps )  =  B )
 
Theoremriota5OLD 6285* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (New usage is discouraged.)
 |-  ( ( ph  /\  B  e.  A  /\  x  e.  A )  ->  ( ps 
 <->  x  =  B ) )   =>    |-  ( ( ph  /\  B  e.  A )  ->  ( iota_ x  e.  A ps )  =  B )
 
Theoremriotass2 6286* Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
 |-  ( ( ( A 
 C_  B  /\  A. x  e.  A  ( ph  ->  ps ) )  /\  ( E. x  e.  A  ph 
 /\  E! x  e.  B  ps ) )  ->  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  B ps ) )
 
Theoremriotass 6287* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  B ph )
 )
 
Theoremmoriotass 6288* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
 |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E* x  e.  B ph )  ->  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  B ph )
 )
 
Theoremsnriota 6289 A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
 |-  ( E! x  e.  A  ph  ->  { x  e.  A  |  ph }  =  { ( iota_ x  e.  A ph ) }
 )
 
Theoremriotaxfrd 6290* Change the variable  x in the expression for "the unique 
x such that  ps " to another variable  y contained in expression  B. Use reuhypd 4519 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ y C   &    |-  ( ( ph  /\  y  e.  A ) 
 ->  B  e.  A )   &    |-  ( ( ph  /\  ( iota_
 y  e.  A ch )  e.  A )  ->  C  e.  A )   &    |-  ( x  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 y  =  ( iota_ y  e.  A ch )  ->  B  =  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  E! y  e.  A  x  =  B )   =>    |-  ( ( ph  /\  E! x  e.  A  ps )  ->  ( iota_ x  e.  A ps )  =  C )
 
Theoremeusvobj2 6291* Specify the same property in two ways when class  B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  B  e.  _V   =>    |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B  <->  A. y  e.  A  x  =  B )
 )
 
Theoremeusvobj1 6292* Specify the same object in two ways when class  B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
 |-  B  e.  _V   =>    |-  ( E! x E. y  e.  A  x  =  B  ->  (
 iota x E. y  e.  A  x  =  B )  =  ( iota x
 A. y  e.  A  x  =  B )
 )
 
Theoremf1ofveu 6293* There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  B )  ->  E! x  e.  A  ( F `  x )  =  C )
 
Theoremf1ocnvfv3 6294* Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  =  (
 iota_ x  e.  A ( F `  x )  =  C ) )
 
Theoremriotaund 6295* Restricted iota equals the undefined value of its domain of discourse  A when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( -.  E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( Undef `  A )
 )
 
Theoremriotaprc 6296* For proper classes, restricted and unrestricted iota are the same. (Contributed by NM, 15-Sep-2011.)
 |-  ( -.  A  e.  _V 
 ->  ( iota_ x  e.  A ph )  =  ( iota
 x ( x  e.  A  /\  ph )
 ) )
 
Theoremriotassuni 6297* The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( iota_ x  e.  A ph )  C_  ( ~P U. A  u.  U. A )
 
Theoremriotaclbg 6298* Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( A  e.  V  ->  ( E! x  e.  A  ph  <->  ( iota_ x  e.  A ph )  e.  A ) )
 
Theoremriotaclb 6299* Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
 |-  A  e.  _V   =>    |-  ( E! x  e.  A  ph  <->  ( iota_ x  e.  A ph )  e.  A )
 
Theoremriotaundb 6300* Restricted iota equals the undefined value of its domain of discourse  A when not meaningful. (Contributed by NM, 26-Sep-2011.)
 |-  A  e.  _V   =>    |-  ( -.  E! x  e.  A  ph  <->  ( iota_ x  e.  A ph )  =  ( Undef `  A )
 )
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