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Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdisjeq1d 4001* 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 4002* 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 4003* 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 4004* 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 )
 
Theoremnfdisj 4005 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  F/_ y A   &    |-  F/_ y B   =>    |-  F/ yDisj  x  e.  A B
 
Theoremnfdisj1 4006 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- 
 F/ xDisj  x  e.  A B
 
Theoremdisjor 4007* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
 |-  ( i  =  j 
 ->  B  =  C )   =>    |-  (Disj  i  e.  A B  <->  A. i  e.  A  A. j  e.  A  (
 i  =  j  \/  ( B  i^i  C )  =  (/) ) )
 
TheoremdisjmoOLD 4008* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.)
 |-  ( i  =  j 
 ->  B  =  C )   =>    |-  ( A. x E* i
 ( i  e.  A  /\  x  e.  B ) 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
 
Theoremdisjors 4009* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  (Disj  x  e.  A B 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
 
Theoremdisji2 4010* Property of a disjoint collection: if  B ( X )  =  C and  B ( Y )  =  D, and  X  =/=  Y, then  C and  D are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  X  ->  B  =  C )   &    |-  ( x  =  Y  ->  B  =  D )   =>    |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A )  /\  X  =/=  Y )  ->  ( C  i^i  D )  =  (/) )
 
Theoremdisji 4011* Property of a disjoint collection: if  B ( X )  =  C and  B ( Y )  =  D have a common element  Z, then  X  =  Y. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  X  ->  B  =  C )   &    |-  ( x  =  Y  ->  B  =  D )   =>    |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A )  /\  ( Z  e.  C  /\  Z  e.  D ) )  ->  X  =  Y )
 
Theoreminvdisj 4012* 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 )
 
Theoremdisjiun 4013* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
 |-  ( (Disj  x  e.  A B  /\  ( C 
 C_  A  /\  D  C_  A  /\  ( C  i^i  D )  =  (/) ) )  ->  ( U_ x  e.  C  B  i^i  U_ x  e.  D  B )  =  (/) )
 
TheoremdisjiunOLD 4014* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.)
 |-  ( ( A. y E* x ( x  e.  A  /\  y  e.  B )  /\  ( C  C_  A  /\  D  C_  A  /\  ( C  i^i  D )  =  (/) ) )  ->  ( U_ x  e.  C  B  i^i  U_ x  e.  D  B )  =  (/) )
 
Theoremsndisj 4015 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  A { x }
 
Theorem0disj 4016 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  A (/)
 
Theoremdisjxsn 4017* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  { A } B
 
Theoremdisjx0 4018 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  (/) B
 
Theoremdisjprg 4019* A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  A  ->  C  =  D )   &    |-  ( x  =  B  ->  C  =  E )   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B ) 
 ->  (Disj  x  e.  { A ,  B } C  <->  ( D  i^i  E )  =  (/) ) )
 
Theoremdisjxiun 4020* An indexed union of a disjoint collection of disjoint collections is disjoint if each component is disjoint, and the disjoint unions in the collection are also disjoint. Note that  B ( y ) and  C
( x ) may have the displayed free variables. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  (Disj  y  e.  A B  ->  (Disj  x  e.  U_ y  e.  A  B C  <->  ( A. y  e.  A Disj  x  e.  B C  /\ Disj  y  e.  A U_ x  e.  B  C ) ) )
 
Theoremdisjxun 4021* The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  y 
 ->  C  =  D )   =>    |-  ( ( A  i^i  B )  =  (/)  ->  (Disj  x  e.  ( A  u.  B ) C  <->  (Disj  x  e.  A C  /\ Disj  x  e.  B C  /\  A. x  e.  A  A. y  e.  B  ( C  i^i  D )  =  (/) ) ) )
 
Theoremdisjss3 4022* Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
 |-  ( ( A  C_  B  /\  A. x  e.  ( B  \  A ) C  =  (/) )  ->  (Disj  x  e.  A C  <-> Disj  x  e.  B C ) )
 
2.1.22  Binary relations
 
Syntaxwbr 4023 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. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 9033.)
 wff  A R B
 
Definitiondf-br 4024 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. Class  R often denotes a relation such as " < " that compares two classes  A and 
B, which might be numbers such as  1 and  2 (see df-ltxr 8867 for the specific definition of  <). As a wff, relations are true or false. For example,  ( R  =  { <. 2 ,  6
>. ,  <. 3 ,  9 >. }  ->  3 R 9 ) (ex-br 20794). Often class  R meets the  Rel criteria to be defined in df-rel 4694, and in particular  R may be a function (see df-fun 5222). 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. (Contributed by NM, 31-Dec-1993.)
 |-  ( A R B  <->  <. A ,  B >.  e.  R )
 
Theorembreq 4025 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
 |-  ( R  =  S  ->  ( A R B  <->  A S B ) )
 
Theorembreq1 4026 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 |-  ( A  =  B  ->  ( A R C  <->  B R C ) )
 
Theorembreq2 4027 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 |-  ( A  =  B  ->  ( C R A  <->  C R B ) )
 
Theorembreq12 4028 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C 
 <->  B R D ) )
 
Theorembreqi 4029 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
 |-  R  =  S   =>    |-  ( A R B 
 <->  A S B )
 
Theorembreq1i 4030 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  A  =  B   =>    |-  ( A R C 
 <->  B R C )
 
Theorembreq2i 4031 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  A  =  B   =>    |-  ( C R A 
 <->  C R B )
 
Theorembreq12i 4032 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 4033 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A R C  <->  B R C ) )
 
Theorembreqd 4034 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C A D  <->  C B D ) )
 
Theorembreq2d 4035 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C R A  <->  C R B ) )
 
Theorembreq12d 4036 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 4037 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 4038 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 4039 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 4040 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 4041 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 4042 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  B R C   =>    |-  A R C
 
Theoremeqbrtrd 4043 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 4044 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  A R C   =>    |-  B R C
 
Theoremeqbrtrrd 4045 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 4046 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A R B   &    |-  B  =  C   =>    |-  A R C
 
Theorembreqtrd 4047 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 4048 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A R B   &    |-  C  =  B   =>    |-  A R C
 
Theorembreqtrrd 4049 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 4050 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 4051 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 4052 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 4053 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 4054 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 4055 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 4056 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 4057 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 4058 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 4059 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 4060 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 4061 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 4062 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 4063 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 4064 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 4065 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 4066 Deduction version of bound-variable hypothesis builder nfbr 4067. (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 4067 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 4068* 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 4069 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
 |-  ( A ( R  u.  S ) B  <-> 
 ( A R B  \/  A S B ) )
 
Theorembrin 4070 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
 |-  ( A ( R  i^i  S ) B  <-> 
 ( A R B  /\  A S B ) )
 
Theorembrdif 4071 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
 |-  ( A ( R 
 \  S ) B  <-> 
 ( A R B  /\  -.  A S B ) )
 
Theoremsbcbrg 4072 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 4073* 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 4074* 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 4075* 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 4076 Extend class notation to include ordered-pair class abstraction (class builder).
 class  { <. x ,  y >.  |  ph }
 
Syntaxcmpt 4077 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 4078* 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 (see dfid2 4309 for a case where they are not distinct). The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. An alternate definition using no existential quantifiers is shown by dfopab2 6135. For example,  R  =  { <. x ,  y
>.  |  ( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1 )  =  y ) }  ->  3 R 4 (ex-opab 20795). (Contributed by NM, 4-Jul-1994.)
 |- 
 { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y ( z  = 
 <. x ,  y >.  /\  ph ) }
 
Definitiondf-mpt 4079* 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. An example is the square function for complex numbers,  ( x  e.  CC  |->  ( x ^ 2 ) ). 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 4080* 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 4081 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 4082* 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 4083 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
 |-  ( ph  <->  ps )   =>    |- 
 { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps }
 
Theoremnfopab 4084* 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 4085 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 4086 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 4087* 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 4088* 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 4089* 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 4090* 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 4091* 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 4092* 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 4093* 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 4094* 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 4095 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 4096 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 4097* 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 4098* 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 4099* 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 4100* 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 ) )
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