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Theorem List for Intuitionistic Logic Explorer - 3801-3900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
2.1.21  Disjointness
 
Syntaxwdisj 3801 Extend wff notation to include the statement that a family of classes  B (
x ), for  x  e.  A, is a disjoint family.
 wff Disj 
 x  e.  A  B
 
Definitiondf-disj 3802* A collection of classes  B ( x ) is disjoint when for each element  y, it is in  B ( x ) for at most one  x. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
 |-  (Disj  x  e.  A  B 
 <-> 
 A. y E* x  e.  A  y  e.  B )
 
Theoremdfdisj2 3803* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
 |-  (Disj  x  e.  A  B 
 <-> 
 A. y E* x ( x  e.  A  /\  y  e.  B ) )
 
Theoremdisjss2 3804 If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A  C  -> Disj  x  e.  A  B ) )
 
Theoremdisjeq2 3805 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( A. x  e.  A  B  =  C  ->  (Disj  x  e.  A  B 
 <-> Disj  x  e.  A  C ) )
 
Theoremdisjeq2dv 3806* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  (Disj  x  e.  A  B  <-> Disj  x  e.  A  C ) )
 
Theoremdisjss1 3807* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( A  C_  B  ->  (Disj  x  e.  B  C  -> Disj  x  e.  A  C ) )
 
Theoremdisjeq1 3808* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( A  =  B  ->  (Disj  x  e.  A  C 
 <-> Disj  x  e.  B  C ) )
 
Theoremdisjeq1d 3809* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  (Disj  x  e.  A  C  <-> Disj  x  e.  B  C ) )
 
Theoremdisjeq12d 3810* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  (Disj  x  e.  A  C  <-> Disj  x  e.  B  D ) )
 
Theoremcbvdisj 3811* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y  ->  B  =  C )   =>    |-  (Disj  x  e.  A  B  <-> Disj  y  e.  A  C )
 
Theoremcbvdisjv 3812* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
 |-  ( x  =  y 
 ->  B  =  C )   =>    |-  (Disj  x  e.  A  B  <-> Disj  y  e.  A  C )
 
Theoremnfdisjv 3813* Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
 |-  F/_ y A   &    |-  F/_ y B   =>    |-  F/ yDisj  x  e.  A  B
 
Theoremnfdisj1 3814 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- 
 F/ xDisj  x  e.  A  B
 
Theoreminvdisj 3815* If there is a function  C ( y ) such that  C ( y )  =  x for all  y  e.  B
( x ), then the sets  B ( x ) for distinct  x  e.  A are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
 |-  ( A. x  e.  A  A. y  e.  B  C  =  x 
 -> Disj 
 x  e.  A  B )
 
Theoremsndisj 3816 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  A  { x }
 
Theorem0disj 3817 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  A  (/)
 
Theoremdisjxsn 3818* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  { A } B
 
Theoremdisjx0 3819 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  (/)  B
 
2.1.22  Binary relations
 
Syntaxwbr 3820 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 3821 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 4669). (Contributed by NM, 31-Dec-1993.)
 |-  ( A R B  <->  <. A ,  B >.  e.  R )
 
Theorembreq 3822 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
 |-  ( R  =  S  ->  ( A R B  <->  A S B ) )
 
Theorembreq1 3823 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 |-  ( A  =  B  ->  ( A R C  <->  B R C ) )
 
Theorembreq2 3824 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 |-  ( A  =  B  ->  ( C R A  <->  C R B ) )
 
Theorembreq12 3825 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C 
 <->  B R D ) )
 
Theorembreqi 3826 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
 |-  R  =  S   =>    |-  ( A R B 
 <->  A S B )
 
Theorembreq1i 3827 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  A  =  B   =>    |-  ( A R C 
 <->  B R C )
 
Theorembreq2i 3828 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  A  =  B   =>    |-  ( C R A 
 <->  C R B )
 
Theorembreq12i 3829 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 3830 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A R C  <->  B R C ) )
 
Theorembreqd 3831 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C A D  <->  C B D ) )
 
Theorembreq2d 3832 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C R A  <->  C R B ) )
 
Theorembreq12d 3833 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 3834 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 ) )
 
Theorembreqan12d 3835 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 3836 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 ) )
 
Theoremnbrne1 3837 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 3838 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 3839 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  B R C   =>    |-  A R C
 
Theoremeqbrtrd 3840 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 3841 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  A R C   =>    |-  B R C
 
Theoremeqbrtrrd 3842 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 3843 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A R B   &    |-  B  =  C   =>    |-  A R C
 
Theorembreqtrd 3844 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 3845 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A R B   &    |-  C  =  B   =>    |-  A R C
 
Theorembreqtrrd 3846 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 3847 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 3848 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 3849 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 3850 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 3851 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 3852 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 )
 
Theoremsyl5eqbr 3853 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 |-  A  =  B   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl5eqbrr 3854 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
 |-  B  =  A   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl5breq 3855 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 |-  A R B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl5breqr 3856 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
 |-  A R B   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl6eqbr 3857 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  B R C   =>    |-  ( ph  ->  A R C )
 
Theoremsyl6eqbrr 3858 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
 |-  ( ph  ->  B  =  A )   &    |-  B R C   =>    |-  ( ph  ->  A R C )
 
Theoremsyl6breq 3859 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 |-  ( ph  ->  A R B )   &    |-  B  =  C   =>    |-  ( ph  ->  A R C )
 
Theoremsyl6breqr 3860 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 3861 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 3862 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 3863 Deduction version of bound-variable hypothesis builder nfbr 3864. (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 3864 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 3865* 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 }
 )
 
Theorembrun 3866 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
 |-  ( A ( R  u.  S ) B  <-> 
 ( A R B  \/  A S B ) )
 
Theorembrin 3867 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
 |-  ( A ( R  i^i  S ) B  <-> 
 ( A R B  /\  A S B ) )
 
Theorembrdif 3868 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
 |-  ( A ( R 
 \  S ) B  <-> 
 ( A R B  /\  -.  A S B ) )
 
Theoremsbcbrg 3869 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 3870* 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 3871* 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 3872* 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 ) )
 
2.1.23  Ordered-pair class abstractions (class builders)
 
Syntaxcopab 3873 Extend class notation to include ordered-pair class abstraction (class builder).
 class  { <. x ,  y >.  |  ph }
 
Syntaxcmpt 3874 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 3875* 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 3876* 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 3877* 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 3878 Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (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 3879* Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  {
 <. x ,  y >.  |  ps }  =  { <. x ,  y >.  |  ch } )
 
Theoremopabbii 3880 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
 |-  ( ph  <->  ps )   =>    |- 
 { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps }
 
Theoremnfopab 3881* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)
 |- 
 F/ z ph   =>    |-  F/_ z { <. x ,  y >.  |  ph }
 
Theoremnfopab1 3882 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 3883 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 3884* 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 3885* 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 3886* 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 3887* 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 3888* 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 3889* 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 3890* 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 3891* 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 3892 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 3893 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 3894* 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 3895* 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 3896* 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 3897* 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 3898* 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 3899 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 3900 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 )
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