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Theorem List for Metamath Proof Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtpostpos 6501 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 6502 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 6503 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
 
Theoremtposfo2 6504 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A -onto-> B  -> tpos 
 F : `' A -onto-> B ) )
 
Theoremtposf2 6505 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )
 
Theoremtposf12 6506 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 6507 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 6508 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 6509 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 6510 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( F  Fn  ( A  X.  B )  -> tpos  F  Fn  ( B  X.  A ) )
 
Theoremtpos0 6511 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
 |- tpos  (/) 
 =  (/)
 
Theoremtposco 6512 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos 
 ( F  o.  G )  =  ( F  o. tpos  G )
 
Theoremtpossym 6513* 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 6514 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  G   =>    |- tpos  F  = tpos  G
 
Theoremtposex 6515 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  e.  _V   =>    |- tpos  F  e.  _V
 
Theoremnftpos 6516 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F/_ x F   =>    |-  F/_ xtpos  F
 
Theoremtposoprab 6517* 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 6518* 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 )
 
2.4.16  Curry and uncurry
 
Syntaxccur 6519 Extend class notation to include the currying function.
 class curry  A
 
Syntaxcunc 6520 Extend class notation to include the uncurrying function.
 class uncurry  A
 
Definitiondf-cur 6521* 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 6522* 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 }
 
2.4.17  Proper subset relation
 
Syntaxcrpss 6523 Extend class notation to include the reified proper subset relation.
 class [ C.]
 
Definitiondf-rpss 6524* Define a relation which corresponds to proper subsethood df-pss 3338 on sets. This allows us to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 6529. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- [ C.]  =  { <. x ,  y >.  |  x  C.  y }
 
Theoremrelrpss 6525 The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- 
 Rel [ C.]
 
Theorembrrpssg 6526 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 6527 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 6528 Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |- [ C.]  Po  A
 
Theoremsorpss 6529* 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 6530 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 6531 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 6532 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 6533* 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 6534* 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 6535* 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 } )
 
2.4.18  Iota properties
 
Theoremfvopab5 6536* 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 6537* 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 6538* 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 6539* 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 6540* 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 )
 )
 
2.4.19  Cantor's Theorem
 
Theoremcanth 6541 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 7262. Note that  A must be a set: this theorem does not hold when  A is too large to be a set; see ncanth 6542 for a counterexample. (Use nex 1565 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 6542 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4343). Specifically, the identity function maps the universe onto its power class. Compare canth 6541 that works for sets. See also the remark in ru 3162 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
 
2.4.20  Undefined values and restricted iota (description binder)
 
Syntaxcund 6543 Extend class notation with undefined value function.
 class  Undef
 
Syntaxcrio 6544 Extend class notation with restricted description binder.
 class  ( iota_ x  e.  A ph )
 
Definitiondf-undef 6545 Define the undefined value function, whose value at set  s is guaranteed not to be a member of 
s (see pwuninel 6547). (Contributed by NM, 15-Sep-2011.)
 |- 
 Undef  =  ( s  e.  _V  |->  ~P U. s )
 
Theorempwuninel2 6546 Direct proof of pwuninel 6547 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 6547 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 6546. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |- 
 -.  ~P U. A  e.  A
 
Theoremundefval 6548 Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 6550 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
 
Theoremundefnel2 6549 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 6550 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 6551 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 5420. (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 6552* 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 6553* 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 6554* 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 6555 Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)
 |-  ( iota_ x  e.  A ps )  e.  _V
 
Theoremriotav 6556 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
 |-  ( iota_ x  e.  _V ph )  =  ( iota
 x ph )
 
Theoremriotaiota 6557 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 6558 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 6559* 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 6560 Deduction version of nfriota 6561. (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 6561* 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 6562* 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 6563* 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 6564* 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 6565 Membership law for "the unique element in  A such that  ph."

This can useful for expanding an iota-based definition (see df-iota 5420). If you have an unbounded iota, iotacl 5443 may be useful.

(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 6566* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
 |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  A )
 
Theoremriotasbc 6567 Substitution law for descriptions. Compare iotasbc 27598. (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 6568* Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2949 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 6569 Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2948 analog.) (Contributed by NM, 16-Jan-2012.)
 |-  ( x  e.  A  ->  ( ph  <->  ps ) )   =>    |-  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  A ps )
 
Theoremriota1 6570* Property of restricted iota. Compare iota1 5434. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota_ x  e.  A ph )  =  x ) )
 
Theoremriota1a 6571 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 6572* A deduction version of riota2f 6573. (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 6573* 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 6574* 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 6575* Properties of a restricted definite description operator. Todo: can some uses of riota2f 6573 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 6576* 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 6577* 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 6578* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( ph  /\  B  e.  A  /\  x  e.  A )  ->  ( ps 
 <->  x  =  B ) )   =>    |-  ( ( ph  /\  B  e.  A )  ->  ( iota_ x  e.  A ps )  =  B )
 
Theoremriotass2 6579* 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 6580* 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 6581* 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 6582 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 6583* Change the variable  x in the expression for "the unique 
x such that  ps " to another variable  y contained in expression  B. Use reuhypd 4752 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 6584* 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 6585* 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 6586* 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 6587* 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 6588* 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 6589* 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 6590* 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 6591* 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 6592* 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 6593* 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 )
 )
 
Theoremriotasvd 6594* Deduction version of riotasv 6599. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ph  ->  D  e.  A )   =>    |-  ( ( ph  /\  A  e.  V )  ->  (
 ( y  e.  B  /\  ps )  ->  D  =  C ) )
 
TheoremriotasvdOLD 6595* Deduction version of riotasv 6599. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  D  =  (
 iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   =>    |-  ( ( ( ph  /\  A  e.  V ) 
 /\  D  e.  A  /\  ( y  e.  B  /\  ps ) )  ->  D  =  C )
 
Theoremriotasv2d 6596* Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4731). Special case of riota2f 6573. (Contributed by NM, 2-Mar-2013.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ y F )   &    |-  ( ph  ->  F/ y ch )   &    |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ( ph  /\  y  =  E ) 
 ->  ( ps  <->  ch ) )   &    |-  (
 ( ph  /\  y  =  E )  ->  C  =  F )   &    |-  ( ph  ->  D  e.  A )   &    |-  ( ph  ->  E  e.  B )   &    |-  ( ph  ->  ch )   =>    |-  (
 ( ph  /\  A  e.  V )  ->  D  =  F )
 
Theoremriotasv2dOLD 6597* Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4731). Special case of riota2f 6573. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  ( z  e.  F  ->  A. y  z  e.  F ) )   &    |-  ( ph  ->  ( ch  ->  A. y ch )
 )   &    |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ph  ->  ( y  =  E  ->  ( ps  <->  ch ) ) )   &    |-  ( ph  ->  ( y  =  E  ->  C  =  F ) )   =>    |-  ( ( (
 ph  /\  A  e.  V )  /\  ( D  e.  A  /\  E  e.  B  /\  ch )
 )  ->  D  =  F )
 
Theoremriotasv2s 6598* The value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4731) in the form of a substitution instance. Special case of riota2f 6573. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  D  =  ( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) )   =>    |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  =  [_ E  /  y ]_ C )
 
Theoremriotasv 6599* Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4731). Special case of riota2f 6573. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  A  e.  _V   &    |-  D  =  ( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) )   =>    |-  ( ( D  e.  A  /\  y  e.  B  /\  ph )  ->  D  =  C )
 
Theoremriotasv3d 6600* A property  ch holding for a representative of a single-valued class expression  C ( y ) (see e.g. reusv2 4731) also holds for its description binder  D (in the form of property  th). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ y th )   &    |-  ( ph  ->  D  =  (
 iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ( ph  /\  C  =  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( ph  ->  ( ( y  e.  B  /\  ps )  ->  ch ) )   &    |-  ( ph  ->  D  e.  A )   &    |-  ( ph  ->  E. y  e.  B  ps )   =>    |-  ( ( ph  /\  A  e.  V ) 
 ->  th )
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