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Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeqrelriiv 4601* Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)
 |- 
 Rel  A   &    |-  Rel  B   &    |-  ( <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B )   =>    |-  A  =  B
 
Theoremeqbrriv 4602* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
 |- 
 Rel  A   &    |-  Rel  B   &    |-  ( x A y  <->  x B y )   =>    |-  A  =  B
 
Theoremeqrelrdv 4603* Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)
 |- 
 Rel  A   &    |-  Rel  B   &    |-  ( ph  ->  (
 <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqbrrdv 4604* Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  Rel  B )   &    |-  ( ph  ->  ( x A y  <->  x B y ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqbrrdiv 4605* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)
 |- 
 Rel  A   &    |-  Rel  B   &    |-  ( ph  ->  ( x A y  <->  x B y ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqrelrdv2 4606* A version of eqrelrdv 4603. (Contributed by Rodolfo Medina, 10-Oct-2010.)
 |-  ( ph  ->  ( <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B ) )   =>    |-  ( ( ( Rel 
 A  /\  Rel  B ) 
 /\  ph )  ->  A  =  B )
 
Theoremssrelrel 4607* A subclass relationship determined by ordered triples. Use relrelss 5033 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  C_  (
 ( _V  X.  _V )  X.  _V )  ->  ( A  C_  B  <->  A. x A. y A. z ( <. <. x ,  y >. ,  z >.  e.  A  ->  <. <. x ,  y >. ,  z >.  e.  B ) ) )
 
Theoremeqrelrel 4608* Extensionality principle for ordered triples, analogous to eqrel 4596. Use relrelss 5033 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)
 |-  ( ( A  u.  B )  C_  ( ( _V  X.  _V )  X.  _V )  ->  ( A  =  B  <->  A. x A. y A. z ( <. <. x ,  y >. ,  z >.  e.  A  <->  <. <. x ,  y >. ,  z >.  e.  B ) ) )
 
Theoremelrel 4609* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
 |-  ( ( Rel  R  /\  A  e.  R ) 
 ->  E. x E. y  A  =  <. x ,  y >. )
 
Theoremrelsng 4610 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)
 |-  ( A  e.  V  ->  ( Rel  { A } 
 <->  A  e.  ( _V 
 X.  _V ) ) )
 
Theoremrelsnopg 4611 A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by BJ, 12-Feb-2022.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  Rel  { <. A ,  B >. } )
 
Theoremrelsn 4612 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)
 |-  A  e.  _V   =>    |-  ( Rel  { A } 
 <->  A  e.  ( _V 
 X.  _V ) )
 
Theoremrelsnop 4613 A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 Rel  { <. A ,  B >. }
 
Theoremxpss12 4614 Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  C_  B  /\  C  C_  D )  ->  ( A  X.  C )  C_  ( B  X.  D ) )
 
Theoremxpss 4615 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
 |-  ( A  X.  B )  C_  ( _V  X.  _V )
 
Theoremrelxp 4616 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)
 |- 
 Rel  ( A  X.  B )
 
Theoremxpss1 4617 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( A  C_  B  ->  ( A  X.  C )  C_  ( B  X.  C ) )
 
Theoremxpss2 4618 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( A  C_  B  ->  ( C  X.  A )  C_  ( C  X.  B ) )
 
Theoremxpsspw 4619 A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)
 |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
 
Theoremunixpss 4620 The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)
 |- 
 U. U. ( A  X.  B )  C_  ( A  u.  B )
 
Theoremxpexg 4621 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B )  e.  _V )
 
Theoremxpex 4622 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  X.  B )  e.  _V
 
Theoremsqxpexg 4623 The Cartesian square of a set is a set. (Contributed by AV, 13-Jan-2020.)
 |-  ( A  e.  V  ->  ( A  X.  A )  e.  _V )
 
Theoremrelun 4624 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)
 |-  ( Rel  ( A  u.  B )  <->  ( Rel  A  /\  Rel  B ) )
 
Theoremrelin1 4625 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
 |-  ( Rel  A  ->  Rel  ( A  i^i  B ) )
 
Theoremrelin2 4626 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
 |-  ( Rel  B  ->  Rel  ( A  i^i  B ) )
 
Theoremreldif 4627 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
 |-  ( Rel  A  ->  Rel  ( A  \  B ) )
 
Theoremreliun 4628 An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)
 |-  ( Rel  U_ x  e.  A  B  <->  A. x  e.  A  Rel  B )
 
Theoremreliin 4629 An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)
 |-  ( E. x  e.  A  Rel  B  ->  Rel  |^|_ x  e.  A  B )
 
Theoremreluni 4630* The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.)
 |-  ( Rel  U. A  <->  A. x  e.  A  Rel  x )
 
Theoremrelint 4631* The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)
 |-  ( E. x  e.  A  Rel  x  ->  Rel  |^| A )
 
Theoremrel0 4632 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
 |- 
 Rel  (/)
 
Theoremrelopabi 4633 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)
 |-  A  =  { <. x ,  y >.  |  ph }   =>    |-  Rel 
 A
 
Theoremrelopab 4634 A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) (Unnecessary distinct variable restrictions were removed by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)
 |- 
 Rel  { <. x ,  y >.  |  ph }
 
Theoremmptrel 4635 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
 |- 
 Rel  ( x  e.  A  |->  B )
 
Theoremreli 4636 The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |- 
 Rel  _I
 
Theoremrele 4637 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |- 
 Rel  _E
 
Theoremopabid2 4638* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
 |-  ( Rel  A  ->  {
 <. x ,  y >.  | 
 <. x ,  y >.  e.  A }  =  A )
 
Theoreminopab 4639* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
 |-  ( { <. x ,  y >.  |  ph }  i^i  {
 <. x ,  y >.  |  ps } )  =  { <. x ,  y >.  |  ( ph  /\  ps ) }
 
Theoremdifopab 4640* The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( { <. x ,  y >.  |  ph }  \  { <. x ,  y >.  |  ps } )  =  { <. x ,  y >.  |  ( ph  /\  -.  ps ) }
 
Theoreminxp 4641 The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  X.  B )  i^i  ( C  X.  D ) )  =  ( ( A  i^i  C )  X.  ( B  i^i  D ) )
 
Theoremxpindi 4642 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
 |-  ( A  X.  ( B  i^i  C ) )  =  ( ( A  X.  B )  i^i  ( A  X.  C ) )
 
Theoremxpindir 4643 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
 |-  ( ( A  i^i  B )  X.  C )  =  ( ( A  X.  C )  i^i  ( B  X.  C ) )
 
Theoremxpiindim 4644* Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
 |-  ( E. y  y  e.  A  ->  ( C  X.  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  ( C  X.  B ) )
 
Theoremxpriindim 4645* Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
 |-  ( E. y  y  e.  A  ->  ( C  X.  ( D  i^i  |^|_
 x  e.  A  B ) )  =  (
 ( C  X.  D )  i^i  |^|_ x  e.  A  ( C  X.  B ) ) )
 
Theoremeliunxp 4646* Membership in a union of cross products. Analogue of elxp 4524 for nonconstant  B ( x ). (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( C  e.  U_ x  e.  A  ( { x }  X.  B ) 
 <-> 
 E. x E. y
 ( C  =  <. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B )
 ) )
 
Theoremopeliunxp2 4647* Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)
 |-  ( x  =  C  ->  B  =  E )   =>    |-  ( <. C ,  D >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) )
 
Theoremraliunxp 4648* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4650, 
B ( y ) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( x  =  <. y ,  z >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  e.  U_  y  e.  A  ( { y }  X.  B ) ph  <->  A. y  e.  A  A. z  e.  B  ps )
 
Theoremrexiunxp 4649* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4651, 
B ( y ) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)
 |-  ( x  =  <. y ,  z >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x  e.  U_  y  e.  A  ( { y }  X.  B ) ph  <->  E. y  e.  A  E. z  e.  B  ps )
 
Theoremralxp 4650* Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)
 |-  ( x  =  <. y ,  z >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. y  e.  A  A. z  e.  B  ps )
 
Theoremrexxp 4651* Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)
 |-  ( x  =  <. y ,  z >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x  e.  ( A  X.  B ) ph  <->  E. y  e.  A  E. z  e.  B  ps )
 
Theoremdjussxp 4652* Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
 |-  U_ x  e.  A  ( { x }  X.  B )  C_  ( A  X.  _V )
 
Theoremralxpf 4653* Version of ralxp 4650 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ z ph   &    |-  F/ x ps   &    |-  ( x  = 
 <. y ,  z >.  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. y  e.  A  A. z  e.  B  ps )
 
Theoremrexxpf 4654* Version of rexxp 4651 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ z ph   &    |-  F/ x ps   &    |-  ( x  = 
 <. y ,  z >.  ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  ( A  X.  B ) ph  <->  E. y  e.  A  E. z  e.  B  ps )
 
Theoremiunxpf 4655* Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)
 |-  F/_ y C   &    |-  F/_ z C   &    |-  F/_ x D   &    |-  ( x  =  <. y ,  z >.  ->  C  =  D )   =>    |-  U_ x  e.  ( A  X.  B ) C  =  U_ y  e.  A  U_ z  e.  B  D
 
Theoremopabbi2dv 4656* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2234. (Contributed by NM, 24-Feb-2014.)
 |- 
 Rel  A   &    |-  ( ph  ->  (
 <. x ,  y >.  e.  A  <->  ps ) )   =>    |-  ( ph  ->  A  =  { <. x ,  y >.  |  ps }
 )
 
Theoremrelop 4657* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( Rel  <. A ,  B >. 
 <-> 
 E. x E. y
 ( A  =  { x }  /\  B  =  { x ,  y }
 ) )
 
Theoremideqg 4658 For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
 
Theoremideq 4659 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
 |-  B  e.  _V   =>    |-  ( A  _I  B 
 <->  A  =  B )
 
Theoremididg 4660 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  e.  V  ->  A  _I  A )
 
Theoremissetid 4661 Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  _V  <->  A  _I  A )
 
Theoremcoss1 4662 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
 |-  ( A  C_  B  ->  ( A  o.  C )  C_  ( B  o.  C ) )
 
Theoremcoss2 4663 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
 |-  ( A  C_  B  ->  ( C  o.  A )  C_  ( C  o.  B ) )
 
Theoremcoeq1 4664 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
 |-  ( A  =  B  ->  ( A  o.  C )  =  ( B  o.  C ) )
 
Theoremcoeq2 4665 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
 |-  ( A  =  B  ->  ( C  o.  A )  =  ( C  o.  B ) )
 
Theoremcoeq1i 4666 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  A  =  B   =>    |-  ( A  o.  C )  =  ( B  o.  C )
 
Theoremcoeq2i 4667 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  A  =  B   =>    |-  ( C  o.  A )  =  ( C  o.  B )
 
Theoremcoeq1d 4668 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  o.  C )  =  ( B  o.  C ) )
 
Theoremcoeq2d 4669 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  o.  A )  =  ( C  o.  B ) )
 
Theoremcoeq12i 4670 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  o.  C )  =  ( B  o.  D )
 
Theoremcoeq12d 4671 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  o.  C )  =  ( B  o.  D ) )
 
Theoremnfco 4672 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  o.  B )
 
Theoremelco 4673* Elements of a composed relation. (Contributed by BJ, 10-Jul-2022.)
 |-  ( A  e.  ( R  o.  S )  <->  E. x E. y E. z ( A  =  <. x ,  z >.  /\  ( x S y 
 /\  y R z ) ) )
 
Theorembrcog 4674* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
 
Theoremopelco2g 4675* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  ( C  o.  D )  <->  E. x ( <. A ,  x >.  e.  D  /\  <. x ,  B >.  e.  C ) ) )
 
Theorembrcogw 4676 Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
 |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z )  /\  ( A D X  /\  X C B ) )  ->  A ( C  o.  D ) B )
 
Theoremeqbrrdva 4677* Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
 |-  ( ph  ->  A  C_  ( C  X.  D ) )   &    |-  ( ph  ->  B 
 C_  ( C  X.  D ) )   &    |-  (
 ( ph  /\  x  e.  C  /\  y  e.  D )  ->  ( x A y  <->  x B y ) )   =>    |-  ( ph  ->  A  =  B )
 
Theorembrco 4678* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) )
 
Theoremopelco 4679* Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  ( C  o.  D )  <->  E. x ( A D x  /\  x C B ) )
 
Theoremcnvss 4680 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  `' A  C_  `' B )
 
Theoremcnveq 4681 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
 |-  ( A  =  B  ->  `' A  =  `' B )
 
Theoremcnveqi 4682 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
 |-  A  =  B   =>    |-  `' A  =  `' B
 
Theoremcnveqd 4683 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  `' A  =  `' B )
 
Theoremelcnv 4684* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
 |-  ( A  e.  `' R 
 <-> 
 E. x E. y
 ( A  =  <. x ,  y >.  /\  y R x ) )
 
Theoremelcnv2 4685* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
 |-  ( A  e.  `' R 
 <-> 
 E. x E. y
 ( A  =  <. x ,  y >.  /\  <. y ,  x >.  e.  R ) )
 
Theoremnfcnv 4686 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x `' A
 
Theoremopelcnvg 4687 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )
 
Theorembrcnvg 4688 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B  <->  B R A ) )
 
Theoremopelcnv 4689 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R )
 
Theorembrcnv 4690 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A `' R B 
 <->  B R A )
 
Theoremcsbcnvg 4691 Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
 |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F )
 
Theoremcnvco 4692 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
 
Theoremcnvuni 4693* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
 |-  `' U. A  =  U_ x  e.  A  `' x
 
Theoremdfdm3 4694* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
 |- 
 dom  A  =  { x  |  E. y <. x ,  y >.  e.  A }
 
Theoremdfrn2 4695* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
 |- 
 ran  A  =  {
 y  |  E. x  x A y }
 
Theoremdfrn3 4696* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
 |- 
 ran  A  =  {
 y  |  E. x <. x ,  y >.  e.  A }
 
Theoremelrn2g 4697* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B ) )
 
Theoremelrng 4698* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x  x B A ) )
 
Theoremdfdm4 4699 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
 |- 
 dom  A  =  ran  `' A
 
Theoremdfdmf 4700* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   =>    |-  dom  A  =  { x  |  E. y  x A y }
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