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Theorem List for Intuitionistic Logic Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremop2ndb 4901 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4290 to extract the first member and op2nda 4902 for an alternate version.) (Contributed by NM, 25-Nov-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 |^| |^| |^| `' { <. A ,  B >. }  =  B
 
Theoremop2nda 4902 Extract the second member of an ordered pair. (See op1sta 4899 to extract the first member and op2ndb 4901 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 U. ran  { <. A ,  B >. }  =  B
 
Theoremcnvsng 4903 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  `' { <. A ,  B >. }  =  { <. B ,  A >. } )
 
Theoremopswapg 4904 Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. `' { <. A ,  B >. }  =  <. B ,  A >. )
 
Theoremelxp4 4905 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4906. (Contributed by NM, 17-Feb-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. U. dom  { A } ,  U. ran  { A } >.  /\  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C )
 ) )
 
Theoremelxp5 4906 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4905 when the double intersection does not create class existence problems (caused by int0 3697). (Contributed by NM, 1-Aug-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C ) ) )
 
Theoremcnvresima 4907 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
 |-  ( `' ( F  |`  A ) " B )  =  ( ( `' F " B )  i^i  A )
 
Theoremresdm2 4908 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
 |-  ( A  |`  dom  A )  =  `' `' A
 
Theoremresdmres 4909 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
 |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
 
Theoremimadmres 4910 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
 |-  ( A " dom  ( A  |`  B ) )  =  ( A
 " B )
 
Theoremmptpreima 4911* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( `' F " C )  =  { x  e.  A  |  B  e.  C }
 
Theoremmptiniseg 4912* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( C  e.  V  ->  ( `' F " { C } )  =  { x  e.  A  |  B  =  C } )
 
Theoremdmmpt 4913 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  dom 
 F  =  { x  e.  A  |  B  e.  _V
 }
 
Theoremdmmptss 4914* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  dom 
 F  C_  A
 
Theoremdmmptg 4915* The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
 |-  ( A. x  e.  A  B  e.  V  ->  dom  ( x  e.  A  |->  B )  =  A )
 
Theoremrelco 4916 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
 |- 
 Rel  ( A  o.  B )
 
Theoremdfco2 4917* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
 |-  ( A  o.  B )  =  U_ x  e. 
 _V  ( ( `' B " { x } )  X.  ( A " { x }
 ) )
 
Theoremdfco2a 4918* Generalization of dfco2 4917, where  C can have any value between  dom  A  i^i  ran 
B and  _V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( A  o.  B )  =  U_ x  e.  C  ( ( `' B " { x } )  X.  ( A " { x }
 ) ) )
 
Theoremcoundi 4919 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  o.  ( B  u.  C ) )  =  ( ( A  o.  B )  u.  ( A  o.  C ) )
 
Theoremcoundir 4920 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  u.  B )  o.  C )  =  ( ( A  o.  C )  u.  ( B  o.  C ) )
 
Theoremcores 4921 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ran  B  C_  C  ->  ( ( A  |`  C )  o.  B )  =  ( A  o.  B ) )
 
Theoremresco 4922 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
 |-  ( ( A  o.  B )  |`  C )  =  ( A  o.  ( B  |`  C ) )
 
Theoremimaco 4923 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
 |-  ( ( A  o.  B ) " C )  =  ( A " ( B " C ) )
 
Theoremrnco 4924 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
 |- 
 ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
 
Theoremrnco2 4925 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
 |- 
 ran  ( A  o.  B )  =  ( A " ran  B )
 
Theoremdmco 4926 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
 |- 
 dom  ( A  o.  B )  =  ( `' B " dom  A )
 
Theoremcoiun 4927* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
 |-  ( A  o.  U_ x  e.  C  B )  =  U_ x  e.  C  ( A  o.  B )
 
Theoremcocnvcnv1 4928 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
 |-  ( `' `' A  o.  B )  =  ( A  o.  B )
 
Theoremcocnvcnv2 4929 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
 |-  ( A  o.  `' `' B )  =  ( A  o.  B )
 
Theoremcores2 4930 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
 |-  ( dom  A  C_  C  ->  ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  B ) )
 
Theoremco02 4931 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
 |-  ( A  o.  (/) )  =  (/)
 
Theoremco01 4932 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
 |-  ( (/)  o.  A )  =  (/)
 
Theoremcoi1 4933 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
 |-  ( Rel  A  ->  ( A  o.  _I  )  =  A )
 
Theoremcoi2 4934 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
 |-  ( Rel  A  ->  (  _I  o.  A )  =  A )
 
Theoremcoires1 4935 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
 |-  ( A  o.  (  _I  |`  B ) )  =  ( A  |`  B )
 
Theoremcoass 4936 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
 |-  ( ( A  o.  B )  o.  C )  =  ( A  o.  ( B  o.  C ) )
 
Theoremrelcnvtr 4937 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  R  ->  ( ( R  o.  R )  C_  R  <->  ( `' R  o.  `' R )  C_  `' R ) )
 
Theoremrelssdmrn 4938 A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
 |-  ( Rel  A  ->  A 
 C_  ( dom  A  X.  ran  A ) )
 
Theoremcnvssrndm 4939 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  `' A  C_  ( ran 
 A  X.  dom  A )
 
Theoremcossxp 4940 Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )
 
Theoremcossxp2 4941 The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.)
 |-  ( ph  ->  R  C_  ( A  X.  B ) )   &    |-  ( ph  ->  S 
 C_  ( B  X.  C ) )   =>    |-  ( ph  ->  ( S  o.  R ) 
 C_  ( A  X.  C ) )
 
Theoremcocnvres 4942 Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.)
 |-  ( S  o.  `' R )  =  (
 ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) )
 
Theoremcocnvss 4943 Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.)
 |-  ( S  o.  `' R )  C_  ( ran  ( R  |`  dom  S )  X.  ran  ( S  |` 
 dom  R ) )
 
Theoremrelrelss 4944 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
 |-  ( ( Rel  A  /\  Rel  dom  A )  <->  A 
 C_  ( ( _V 
 X.  _V )  X.  _V ) )
 
Theoremunielrel 4945 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
 |-  ( ( Rel  R  /\  A  e.  R ) 
 ->  U. A  e.  U. R )
 
Theoremrelfld 4946 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
 |-  ( Rel  R  ->  U.
 U. R  =  ( dom  R  u.  ran  R ) )
 
Theoremrelresfld 4947 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
 |-  ( Rel  R  ->  ( R  |`  U. U. R )  =  R )
 
Theoremrelcoi2 4948 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
 |-  ( Rel  R  ->  ( (  _I  |`  U. U. R )  o.  R )  =  R )
 
Theoremrelcoi1 4949 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)
 |-  ( Rel  R  ->  ( R  o.  (  _I  |`  U. U. R ) )  =  R )
 
Theoremunidmrn 4950 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
 |- 
 U. U. `' A  =  ( dom  A  u.  ran  A )
 
Theoremrelcnvfld 4951 if  R is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
 |-  ( Rel  R  ->  U.
 U. R  =  U. U. `' R )
 
Theoremdfdm2 4952 Alternate definition of domain df-dm 4438 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
 |- 
 dom  A  =  U. U. ( `' A  o.  A )
 
Theoremunixpm 4953* The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)
 |-  ( E. x  x  e.  ( A  X.  B )  ->  U. U. ( A  X.  B )  =  ( A  u.  B ) )
 
Theoremunixp0im 4954 The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)
 |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  =  (/) )
 
Theoremcnvexg 4955 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
 |-  ( A  e.  V  ->  `' A  e.  _V )
 
Theoremcnvex 4956 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
 |-  A  e.  _V   =>    |-  `' A  e.  _V
 
Theoremrelcnvexb 4957 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
 |-  ( Rel  R  ->  ( R  e.  _V  <->  `' R  e.  _V ) )
 
Theoremressn 4958 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( A  |`  { B } )  =  ( { B }  X.  ( A " { B }
 ) )
 
Theoremcnviinm 4959* The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
 |-  ( E. y  y  e.  A  ->  `' |^|_ x  e.  A  B  =  |^|_
 x  e.  A  `' B )
 
Theoremcnvpom 4960* The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
 |-  ( E. x  x  e.  A  ->  ( R  Po  A  <->  `' R  Po  A ) )
 
Theoremcnvsom 4961* The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
 |-  ( E. x  x  e.  A  ->  ( R  Or  A  <->  `' R  Or  A ) )
 
Theoremcoexg 4962 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  o.  B )  e.  _V )
 
Theoremcoex 4963 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  o.  B )  e.  _V
 
Theoremxpcom 4964* Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
 |-  ( E. x  x  e.  B  ->  (
 ( B  X.  C )  o.  ( A  X.  B ) )  =  ( A  X.  C ) )
 
2.6.7  Definite description binder (inverted iota)
 
Syntaxcio 4965 Extend class notation with Russell's definition description binder (inverted iota).
 class  ( iota x ph )
 
Theoremiotajust 4966* Soundness justification theorem for df-iota 4967. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |- 
 U. { y  |  { x  |  ph }  =  { y } }  =  U. { z  |  { x  |  ph }  =  { z } }
 
Definitiondf-iota 4967* 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 4978); otherwise, it evaluates to the empty set (see iotanul 4982). 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 iotacl 4990 (for unbounded iota). This can be easier than applying 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 4968* 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 4969 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 )
 
Theoremnfiotadxy 4970* Deduction version of nfiotaxy 4971. (Contributed by Jim Kingdon, 21-Dec-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/_ x ( iota y ps ) )
 
Theoremnfiotaxy 4971* Bound-variable hypothesis builder for the  iota class. (Contributed by NM, 23-Aug-2011.)
 |- 
 F/ x ph   =>    |-  F/_ x ( iota y ph )
 
Theoremcbviota 4972 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 4973* 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 4974 Variable substitution in description binder. Compare sb8eu 1961. (Contributed by NM, 18-Mar-2013.)
 |- 
 F/ y ph   =>    |-  ( iota x ph )  =  ( iota y [ y  /  x ] ph )
 
Theoremiotaeq 4975 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )
 
Theoremiotabi 4976 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )
 
Theoremuniabio 4977* 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 4978* 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 4979 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  =  U. { x  |  ph } )
 
Theoremiotaint 4980 Equivalence between two different forms of  iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E! x ph  ->  ( iota x ph )  =  |^| { x  |  ph } )
 
Theoremiota1 4981 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 4982 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 )  =  (/) )
 
Theoremeuiotaex 4983 Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the  iota class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)
 |-  ( E! x ph  ->  ( iota x ph )  e.  _V )
 
Theoremiotass 4984* Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)
 |-  ( A. x (
 ph  ->  x  C_  A )  ->  ( iota x ph )  C_  A )
 
Theoremiota4 4985 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  -> 
 [. ( iota x ph )  /  x ]. ph )
 
Theoremiota4an 4986 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 4987* 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 4988* Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota x ps )  =  ( iota x ch ) )
 
Theoremiotabii 4989 Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( ph  <->  ps )   =>    |-  ( iota x ph )  =  ( iota x ps )
 
Theoremiotacl 4990 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 4967).

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

 |-  ( E! x ph  ->  ( iota x ph )  e.  { x  |  ph } )
 
Theoremiota2df 4991 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 4992* 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 4993* 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 4994 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 ) } )
 
Theoremcsbiotag 4995* Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ].
 ph ) )
 
2.6.8  Functions
 
Syntaxwfun 4996 Extend the definition of a wff to include the function predicate. (Read:  A is a function.)
 wff  Fun  A
 
Syntaxwfn 4997 Extend the definition of a wff to include the function predicate with a domain. (Read:  A is a function on  B.)
 wff  A  Fn  B
 
Syntaxwf 4998 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: 
F maps  A into  B.)
 wff  F : A --> B
 
Syntaxwf1 4999 Extend the definition of a wff to include one-to-one functions. (Read:  F maps  A one-to-one into  B.) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.
 wff  F : A -1-1-> B
 
Syntaxwfo 5000 Extend the definition of a wff to include onto functions. (Read:  F maps  A onto  B.) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.
 wff  F : A -onto-> B
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