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Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdisjxsn 4001* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  { A } B
 
Theoremdisjx0 4002 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  (/)  B
 
2.1.22  Binary relations
 
Syntaxwbr 4003 Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous.
 wff  A R B
 
Definitiondf-br 4004 Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. This definition of relations is well-defined, although not very meaningful, when classes  A and/or  B are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when  R is a proper class (see for example iprc 4895). (Contributed by NM, 31-Dec-1993.)
 |-  ( A R B  <->  <. A ,  B >.  e.  R )
 
Theorembreq 4005 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
 |-  ( R  =  S  ->  ( A R B  <->  A S B ) )
 
Theorembreq1 4006 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 |-  ( A  =  B  ->  ( A R C  <->  B R C ) )
 
Theorembreq2 4007 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 |-  ( A  =  B  ->  ( C R A  <->  C R B ) )
 
Theorembreq12 4008 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C 
 <->  B R D ) )
 
Theorembreqi 4009 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
 |-  R  =  S   =>    |-  ( A R B 
 <->  A S B )
 
Theorembreq1i 4010 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  A  =  B   =>    |-  ( A R C 
 <->  B R C )
 
Theorembreq2i 4011 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  A  =  B   =>    |-  ( C R A 
 <->  C R B )
 
Theorembreq12i 4012 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A R C  <->  B R D )
 
Theorembreq1d 4013 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A R C  <->  B R C ) )
 
Theorembreqd 4014 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C A D  <->  C B D ) )
 
Theorembreq2d 4015 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C R A  <->  C R B ) )
 
Theorembreq12d 4016 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A R C  <->  B R D ) )
 
Theorembreq123d 4017 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  R  =  S )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A R C  <->  B S D ) )
 
Theorembreqdi 4018 Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C A D )   =>    |-  ( ph  ->  C B D )
 
Theorembreqan12d 4019 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( A R C  <->  B R D ) )
 
Theorembreqan12rd 4020 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ps 
 /\  ph )  ->  ( A R C  <->  B R D ) )
 
Theoremeqnbrtrd 4021 Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  -.  B R C )   =>    |-  ( ph  ->  -.  A R C )
 
Theoremnbrne1 4022 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
 |-  ( ( A R B  /\  -.  A R C )  ->  B  =/=  C )
 
Theoremnbrne2 4023 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
 |-  ( ( A R C  /\  -.  B R C )  ->  A  =/=  B )
 
Theoremeqbrtri 4024 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  B R C   =>    |-  A R C
 
Theoremeqbrtrd 4025 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremeqbrtrri 4026 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  A R C   =>    |-  B R C
 
Theoremeqbrtrrd 4027 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A R C )   =>    |-  ( ph  ->  B R C )
 
Theorembreqtri 4028 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A R B   &    |-  B  =  C   =>    |-  A R C
 
Theorembreqtrd 4029 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
 |-  ( ph  ->  A R B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A R C )
 
Theorembreqtrri 4030 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A R B   &    |-  C  =  B   =>    |-  A R C
 
Theorembreqtrrd 4031 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
 |-  ( ph  ->  A R B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A R C )
 
Theorem3brtr3i 4032 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
 |-  A R B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  C R D
 
Theorem3brtr4i 4033 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
 |-  A R B   &    |-  C  =  A   &    |-  D  =  B   =>    |-  C R D
 
Theorem3brtr3d 4034 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
 |-  ( ph  ->  A R B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  C R D )
 
Theorem3brtr4d 4035 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
 |-  ( ph  ->  A R B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  C R D )
 
Theorem3brtr3g 4036 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
 |-  ( ph  ->  A R B )   &    |-  A  =  C   &    |-  B  =  D   =>    |-  ( ph  ->  C R D )
 
Theorem3brtr4g 4037 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
 |-  ( ph  ->  A R B )   &    |-  C  =  A   &    |-  D  =  B   =>    |-  ( ph  ->  C R D )
 
Theoremeqbrtrid 4038 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 |-  A  =  B   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremeqbrtrrid 4039 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
 |-  B  =  A   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theorembreqtrid 4040 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 |-  A R B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A R C )
 
Theorembreqtrrid 4041 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
 |-  A R B   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A R C )
 
Theoremeqbrtrdi 4042 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  B R C   =>    |-  ( ph  ->  A R C )
 
Theoremeqbrtrrdi 4043 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
 |-  ( ph  ->  B  =  A )   &    |-  B R C   =>    |-  ( ph  ->  A R C )
 
Theorembreqtrdi 4044 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 |-  ( ph  ->  A R B )   &    |-  B  =  C   =>    |-  ( ph  ->  A R C )
 
Theorembreqtrrdi 4045 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
 |-  ( ph  ->  A R B )   &    |-  C  =  B   =>    |-  ( ph  ->  A R C )
 
Theoremssbrd 4046 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( C A D  ->  C B D ) )
 
Theoremssbri 4047 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
 |-  A  C_  B   =>    |-  ( C A D  ->  C B D )
 
Theoremnfbrd 4048 Deduction version of bound-variable hypothesis builder nfbr 4049. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x R )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/ x  A R B )
 
Theoremnfbr 4049 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x R   &    |-  F/_ x B   =>    |- 
 F/ x  A R B
 
Theorembrab1 4050* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
 |-  ( x R A  <->  x  e.  { z  |  z R A }
 )
 
Theorembr0 4051 The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
 |- 
 -.  A (/) B
 
Theorembrne0 4052 If two sets are in a binary relation, the relation cannot be empty. In fact, the relation is also inhabited, as seen at brm 4053. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
 |-  ( A R B  ->  R  =/=  (/) )
 
Theorembrm 4053* If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
 |-  ( A R B  ->  E. x  x  e.  R )
 
Theorembrun 4054 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
 |-  ( A ( R  u.  S ) B  <-> 
 ( A R B  \/  A S B ) )
 
Theorembrin 4055 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
 |-  ( A ( R  i^i  S ) B  <-> 
 ( A R B  /\  A S B ) )
 
Theorembrdif 4056 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
 |-  ( A ( R 
 \  S ) B  <-> 
 ( A R B  /\  -.  A S B ) )
 
Theoremsbcbrg 4057 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C ) )
 
Theoremsbcbr12g 4058* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
 |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R [_ A  /  x ]_ C ) )
 
Theoremsbcbr1g 4059* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
 |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B R C ) )
 
Theoremsbcbr2g 4060* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
 |-  ( A  e.  D  ->  ( [. A  /  x ]. B R C  <->  B R [_ A  /  x ]_ C ) )
 
Theorembrralrspcev 4061* Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.)
 |-  ( ( B  e.  X  /\  A. y  e.  Y  A R B )  ->  E. x  e.  X  A. y  e.  Y  A R x )
 
Theorembrimralrspcev 4062* Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.)
 |-  ( ( B  e.  X  /\  A. y  e.  Y  ( ( ph  /\  A R B ) 
 ->  ps ) )  ->  E. x  e.  X  A. y  e.  Y  ( ( ph  /\  A R x )  ->  ps )
 )
 
2.1.23  Ordered-pair class abstractions (class builders)
 
Syntaxcopab 4063 Extend class notation to include ordered-pair class abstraction (class builder).
 class  { <. x ,  y >.  |  ph }
 
Syntaxcmpt 4064 Extend the definition of a class to include maps-to notation for defining a function via a rule.
 class  ( x  e.  A  |->  B )
 
Definitiondf-opab 4065* Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually  x and  y are distinct, although the definition doesn't strictly require it. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.)
 |- 
 { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y ( z  = 
 <. x ,  y >.  /\  ph ) }
 
Definitiondf-mpt 4066* Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from  x (in 
A) to  B ( x )". The class expression  B is the value of the function at  x and normally contains the variable  x. Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)
 |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
 
Theoremopabss 4067* The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |- 
 { <. x ,  y >.  |  x R y }  C_  R
 
Theoremopabbid 4068 Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |- 
 F/ x ph   &    |-  F/ y ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  {
 <. x ,  y >.  |  ps }  =  { <. x ,  y >.  |  ch } )
 
Theoremopabbidv 4069* Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  {
 <. x ,  y >.  |  ps }  =  { <. x ,  y >.  |  ch } )
 
Theoremopabbii 4070 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
 |-  ( ph  <->  ps )   =>    |- 
 { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps }
 
Theoremnfopab 4071* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.)
 |- 
 F/ z ph   =>    |-  F/_ z { <. x ,  y >.  |  ph }
 
Theoremnfopab1 4072 The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x { <. x ,  y >.  |  ph }
 
Theoremnfopab2 4073 The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ y { <. x ,  y >.  |  ph }
 
Theoremcbvopab 4074* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
 |- 
 F/ z ph   &    |-  F/ w ph   &    |-  F/ x ps   &    |-  F/ y ps   &    |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
 
Theoremcbvopabv 4075* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
 |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  { <. x ,  y >.  |  ph }  =  { <. z ,  w >.  |  ps }
 
Theoremcbvopab1 4076* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |- 
 F/ z ph   &    |-  F/ x ps   &    |-  ( x  =  z  ->  (
 ph 
 <->  ps ) )   =>    |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  ps }
 
Theoremcbvopab2 4077* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
 |- 
 F/ z ph   &    |-  F/ y ps   &    |-  ( y  =  z  ->  ( ph  <->  ps ) )   =>    |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
 
Theoremcbvopab1s 4078* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)
 |- 
 { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  [ z  /  x ] ph }
 
Theoremcbvopab1v 4079* Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
 |-  ( x  =  z 
 ->  ( ph  <->  ps ) )   =>    |-  { <. x ,  y >.  |  ph }  =  { <. z ,  y >.  |  ps }
 
Theoremcbvopab2v 4080* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
 |-  ( y  =  z 
 ->  ( ph  <->  ps ) )   =>    |-  { <. x ,  y >.  |  ph }  =  { <. x ,  z >.  |  ps }
 
Theoremcsbopabg 4081* Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph } )
 
Theoremunopab 4082 Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
 |-  ( { <. x ,  y >.  |  ph }  u.  {
 <. x ,  y >.  |  ps } )  =  { <. x ,  y >.  |  ( ph  \/  ps ) }
 
Theoremmpteq12f 4083 An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
 
Theoremmpteq12dva 4084* An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  A  =  C )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  =  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
 
Theoremmpteq12dv 4085* An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
 |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
 
Theoremmpteq12 4086* An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.)
 |-  ( ( A  =  C  /\  A. x  e.  A  B  =  D )  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
 
Theoremmpteq1 4087* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 |-  ( A  =  B  ->  ( x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
 
Theoremmpteq1d 4088* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
 
Theoremmpteq2ia 4089 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 |-  ( x  e.  A  ->  B  =  C )   =>    |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C )
 
Theoremmpteq2i 4090 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 |-  B  =  C   =>    |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C )
 
Theoremmpteq12i 4091 An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)
 |-  A  =  C   &    |-  B  =  D   =>    |-  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D )
 
Theoremmpteq2da 4092 Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
 
Theoremmpteq2dva 4093* Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
 
Theoremmpteq2dv 4094* An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
 
Theoremnfmpt 4095* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( y  e.  A  |->  B )
 
Theoremnfmpt1 4096 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
 |-  F/_ x ( x  e.  A  |->  B )
 
Theoremcbvmptf 4097* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Thierry Arnoux, 9-Mar-2017.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y  ->  B  =  C )   =>    |-  ( x  e.  A  |->  B )  =  (
 y  e.  A  |->  C )
 
Theoremcbvmpt 4098* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)
 |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y  ->  B  =  C )   =>    |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
 
Theoremcbvmptv 4099* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)
 |-  ( x  =  y 
 ->  B  =  C )   =>    |-  ( x  e.  A  |->  B )  =  (
 y  e.  A  |->  C )
 
Theoremmptv 4100* Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
 |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  y  =  B }
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