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Theorem List for Metamath Proof Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremovtpos 6101 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 6102 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Fun  F  ->  Fun tpos  F )
 
Theoremdftpos2 6103* 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 6104* Alternate definition of tpos when 
F has relational domain. Compare df-cnv 4596. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  -> tpos 
 F  =  { <. <. x ,  y >. ,  z >.  |  <. y ,  x >. F z }
 )
 
Theoremdftpos4 6105* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos  F  =  ( F  o.  ( x  e.  (
 ( _V  X.  _V )  u.  { (/) } )  |-> 
 U. `' { x } ) )
 
Theoremtpostpos 6106 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 6107 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 6108 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
 
Theoremtposfo2 6109 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A -onto-> B  -> tpos 
 F : `' A -onto-> B ) )
 
Theoremtposf2 6110 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )
 
Theoremtposf12 6111 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 6112 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 6113 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 6114 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 6115 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( F  Fn  ( A  X.  B )  -> tpos  F  Fn  ( B  X.  A ) )
 
Theoremtpos0 6116 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
 |- tpos  (/) 
 =  (/)
 
Theoremtposco 6117 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos 
 ( F  o.  G )  =  ( F  o. tpos  G )
 
Theoremtpossym 6118* 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 6119 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  G   =>    |- tpos  F  = tpos  G
 
Theoremtposex 6120 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  e.  _V   =>    |- tpos  F  e.  _V
 
Theoremnftpos 6121 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F/_ x F   =>    |-  F/_ xtpos  F
 
Theoremtposoprab 6122* 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 6123* 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 )
 
2.4.13  Curry and uncurry
 
Syntaxccur 6124 Extend class notation to include the currying function.
 class curry  A
 
Syntaxcunc 6125 Extend class notation to include the uncurrying function.
 class uncurry  A
 
Definitiondf-cur 6126* 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 6127* 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 }
 
2.4.14  Proper subset relation
 
Syntaxcrpss 6128 Extend class notation to include the reified proper subset relation.
 class [ C.]
 
Definitiondf-rpss 6129* Define a relation which corresponds to proper subsethood df-pss 3091 on sets. This allows us to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 6134. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- [ C.]  =  { <. x ,  y >.  |  x  C.  y }
 
Theoremrelrpss 6130 The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- 
 Rel [ C.]
 
Theorembrrpssg 6131 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 6132 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 6133 Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- [ C.]  Po  A
 
Theoremsorpss 6134* 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 6135 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 6136 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 6137 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 6138* 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 6139* 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 6140* 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 6141 Extend class notation with Russell's definition description binder (inverted iota).
 class  ( iota x ph )
 
Theoremiotajust 6142* Soundness justification theorem for df-iota 6143. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |- 
 U. { y  |  { x  |  ph }  =  { y } }  =  U. { z  |  { x  |  ph }  =  { z } }
 
Definitiondf-iota 6143* 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 6154); otherwise, it evaluates to the empty set (see iotanul 6158). 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 6144* 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 6145 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 6146 Deduction version of nfiota 6147. (Contributed by NM, 18-Feb-2013.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/_ x ( iota y ps ) )
 
Theoremnfiota 6147 Bound-variable hypothesis builder for the  iota class. (Contributed by NM, 23-Aug-2011.)
 |- 
 F/ x ph   =>    |-  F/_ x ( iota y ph )
 
Theoremcbviota 6148 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 6149* 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 6150 Variable substitution in description binder. Compare sb8eu 2132. (Contributed by NM, 18-Mar-2013.)
 |- 
 F/ y ph   =>    |-  ( iota x ph )  =  ( iota y [ y  /  x ] ph )
 
Theoremiotaeq 6151 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )
 
Theoremiotabi 6152 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )
 
Theoremuniabio 6153* 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 6154* 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 6155 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  =  U. { x  |  ph } )
 
Theoremiotaint 6156 Equivalence between two different forms of  iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E! x ph  ->  ( iota x ph )  =  |^| { x  |  ph } )
 
Theoremiota1 6157 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 6158 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 6159 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 6160 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 6161 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  -> 
 [. ( iota x ph )  /  x ]. ph )
 
Theoremiota4an 6162 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 6163* 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 6164* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota x ps )  =  ( iota x ch ) )
 
Theoremiotabii 6165 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( ph  <->  ps )   =>    |-  ( iota x ph )  =  ( iota x ps )
 
Theoremiotacl 6166 Membership law for descriptions. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  e.  { x  |  ph } )
 
Theoremiota2df 6167 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 6168* 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 6169* 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 6170 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 6171* 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 6172* 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 6173* 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 6174* 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 6175* 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 6176* 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 6177* 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 6178 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 6899. Note that  A must be a set: this theorem does not hold when  A is too large to be a set; see ncanth 6179 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 6179 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4049). Specifically, the identity function maps the universe onto its power class. Compare canth 6178 that works for sets. See also the remark in ru 2920 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 6180 Extend class notation with undefined value function.
 class  Undef
 
Syntaxcrio 6181 Extend class notation with restricted description binder.
 class  ( iota_ x  e.  A ph )
 
Definitiondf-undef 6182 Define the undefined value function, whose value at set  s is guaranteed not to be a member of 
s (see pwuninel 6186). (Contributed by NM, 15-Sep-2011.)
 |- 
 Undef  =  ( s  e.  _V  |->  ~P U. s )
 
Theorempwnss 6183 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 6184 No set equals its power set. The sethood antecedent is necessary; compare pwv 3726. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  ( A  e.  V  ->  ~P A  =/=  A )
 
TheorempwuninelALT 6185 Direct proof of pwuninel 6186 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 6186 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 6187 Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 6189 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
 
Theoremundefnel2 6188 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 6189 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 6190 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 6143. (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 6191* 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 6192* 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 6193* 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 6194 Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)
 |-  ( iota_ x  e.  A ps )  e.  _V
 
Theoremriotav 6195 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
 |-  ( iota_ x  e.  _V ph )  =  ( iota
 x ph )
 
Theoremriotaiota 6196 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 6197 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 6198* 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 6199 Deduction version of nfriota 6200. (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 6200* 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 )
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