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Theorem List for Intuitionistic Logic Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreseq2d 4701 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  |`  A )  =  ( C  |`  B ) )
 
Theoremreseq12d 4702 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )
 
Theoremnfres 4703 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  |`  B )
 
Theoremcsbresg 4704 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B  |`  C )  =  ( [_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) )
 
Theoremres0 4705 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
 |-  ( A  |`  (/) )  =  (/)
 
Theoremopelres 4706 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
 |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  ( C  |`  D )  <->  ( <. A ,  B >.  e.  C  /\  A  e.  D )
 )
 
Theorembrres 4707 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
 |-  B  e.  _V   =>    |-  ( A ( C  |`  D ) B 
 <->  ( A C B  /\  A  e.  D ) )
 
Theoremopelresg 4708 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)
 |-  ( B  e.  V  ->  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
 ( <. A ,  B >.  e.  C  /\  A  e.  D ) ) )
 
Theorembrresg 4709 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
 |-  ( B  e.  V  ->  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) ) )
 
Theoremopres 4710 Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  B  e.  _V   =>    |-  ( A  e.  D  ->  ( <. A ,  B >.  e.  ( C  |`  D )  <->  <. A ,  B >.  e.  C ) )
 
Theoremresieq 4711 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
 |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B (  _I  |`  A ) C 
 <->  B  =  C ) )
 
Theoremopelresi 4712  <. A ,  A >. belongs to a restriction of the identity class iff  A belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)
 |-  ( A  e.  V  ->  ( <. A ,  A >.  e.  (  _I  |`  B )  <->  A  e.  B )
 )
 
Theoremresres 4713 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
 |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
 
Theoremresundi 4714 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
 |-  ( A  |`  ( B  u.  C ) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
 
Theoremresundir 4715 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
 |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
 
Theoremresindi 4716 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
 |-  ( A  |`  ( B  i^i  C ) )  =  ( ( A  |`  B )  i^i  ( A  |`  C ) )
 
Theoremresindir 4717 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
 |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
 
Theoreminres 4718 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
 |-  ( A  i^i  ( B  |`  C ) )  =  ( ( A  i^i  B )  |`  C )
 
Theoremresiun1 4719* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( U_ x  e.  A  B  |`  C )  =  U_ x  e.  A  ( B  |`  C )
 
Theoremresiun2 4720* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( C  |`  U_ x  e.  A  B )  = 
 U_ x  e.  A  ( C  |`  B )
 
Theoremdmres 4721 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
 |- 
 dom  ( A  |`  B )  =  ( B  i^i  dom 
 A )
 
Theoremssdmres 4722 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
 |-  ( A  C_  dom  B  <->  dom  ( B  |`  A )  =  A )
 
Theoremdmresexg 4723 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
 |-  ( B  e.  V  ->  dom  ( A  |`  B )  e.  _V )
 
Theoremresss 4724 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
 |-  ( A  |`  B ) 
 C_  A
 
Theoremrescom 4725 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
 
Theoremssres 4726 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
 |-  ( A  C_  B  ->  ( A  |`  C ) 
 C_  ( B  |`  C ) )
 
Theoremssres2 4727 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  C_  B  ->  ( C  |`  A ) 
 C_  ( C  |`  B ) )
 
Theoremrelres 4728 A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 Rel  ( A  |`  B )
 
Theoremresabs1 4729 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
 |-  ( B  C_  C  ->  ( ( A  |`  C )  |`  B )  =  ( A  |`  B )
 )
 
Theoremresabs2 4730 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( B  C_  C  ->  ( ( A  |`  B )  |`  C )  =  ( A  |`  B )
 )
 
Theoremresidm 4731 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
 
Theoremresima 4732 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
 |-  ( ( A  |`  B )
 " B )  =  ( A " B )
 
Theoremresima2 4733 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
 |-  ( B  C_  C  ->  ( ( A  |`  C )
 " B )  =  ( A " B ) )
 
Theoremxpssres 4734 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( C  C_  A  ->  ( ( A  X.  B )  |`  C )  =  ( C  X.  B ) )
 
Theoremelres 4735* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
 |-  ( A  e.  ( B  |`  C )  <->  E. x  e.  C  E. y ( A  =  <. x ,  y >.  /\ 
 <. x ,  y >.  e.  B ) )
 
Theoremelsnres 4736* Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)
 |-  C  e.  _V   =>    |-  ( A  e.  ( B  |`  { C } )  <->  E. y ( A  =  <. C ,  y >.  /\  <. C ,  y >.  e.  B ) )
 
Theoremrelssres 4737 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
 |-  ( ( Rel  A  /\  dom  A  C_  B )  ->  ( A  |`  B )  =  A )
 
Theoremresdm 4738 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
 |-  ( Rel  A  ->  ( A  |`  dom  A )  =  A )
 
Theoremresexg 4739 The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  e.  V  ->  ( A  |`  B )  e.  _V )
 
Theoremresex 4740 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  A  e.  _V   =>    |-  ( A  |`  B )  e.  _V
 
Theoremresindm 4741 When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.)
 |-  ( Rel  A  ->  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  B ) )
 
Theoremresdmdfsn 4742 Restricting a relation to its domain without a set is the same as restricting the relation to the universe without this set. (Contributed by AV, 2-Dec-2018.)
 |-  ( Rel  R  ->  ( R  |`  ( _V  \  { X } )
 )  =  ( R  |`  ( dom  R  \  { X } ) ) )
 
Theoremresopab 4743* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
 |-  ( { <. x ,  y >.  |  ph }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
 
Theoremresiexg 4744 The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)
 |-  ( A  e.  V  ->  (  _I  |`  A )  e.  _V )
 
Theoremiss 4745 A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  C_  _I  <->  A  =  (  _I  |`  dom  A )
 )
 
Theoremresopab2 4746* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
 |-  ( A  C_  B  ->  ( { <. x ,  y >.  |  ( x  e.  B  /\  ph ) }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } )
 
Theoremresmpt 4747* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
 |-  ( B  C_  A  ->  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
 
Theoremresmpt3 4748* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
 |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  ( A  i^i  B )  |->  C )
 
Theoremresmptf 4749 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( B  C_  A  ->  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
 
Theoremresmptd 4750* Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  B  C_  A )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
 
Theoremdfres2 4751* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  ( R  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
 
Theoremopabresid 4752* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
 |- 
 { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
 
Theoremmptresid 4753* The restricted identity expressed with the maps-to notation. (Contributed by FL, 25-Apr-2012.)
 |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
 
Theoremdmresi 4754 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 dom  (  _I  |`  A )  =  A
 
Theoremresid 4755 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
 |-  ( Rel  A  ->  ( A  |`  _V )  =  A )
 
Theoremimaeq1 4756 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2 4757 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq1i 4758 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A " C )  =  ( B " C )
 
Theoremimaeq2i 4759 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C " A )  =  ( C " B )
 
Theoremimaeq1d 4760 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2d 4761 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq12d 4762 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A " C )  =  ( B " D ) )
 
Theoremdfima2 4763* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A " B )  =  { y  |  E. x  e.  B  x A y }
 
Theoremdfima3 4764* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A " B )  =  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  A ) }
 
Theoremelimag 4765* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
 |-  ( A  e.  V  ->  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A ) )
 
Theoremelima 4766* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
 |-  A  e.  _V   =>    |-  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A )
 
Theoremelima2 4767* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
 |-  A  e.  _V   =>    |-  ( A  e.  ( B " C )  <->  E. x ( x  e.  C  /\  x B A ) )
 
Theoremelima3 4768* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  ( B " C )  <->  E. x ( x  e.  C  /\  <. x ,  A >.  e.  B ) )
 
Theoremnfima 4769 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A
 " B )
 
Theoremnfimad 4770 Deduction version of bound-variable hypothesis builder nfima 4769. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x ( A " B ) )
 
Theoremimadmrn 4771 The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)
 |-  ( A " dom  A )  =  ran  A
 
Theoremimassrn 4772 The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.)
 |-  ( A " B )  C_  ran  A
 
Theoremimaexg 4773 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
 |-  ( A  e.  V  ->  ( A " B )  e.  _V )
 
Theoremimaex 4774 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.)
 |-  A  e.  _V   =>    |-  ( A " B )  e.  _V
 
Theoremimai 4775 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
 |-  (  _I  " A )  =  A
 
Theoremrnresi 4776 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 ran  (  _I  |`  A )  =  A
 
Theoremresiima 4777 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
 |-  ( B  C_  A  ->  ( (  _I  |`  A )
 " B )  =  B )
 
Theoremima0 4778 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
 |-  ( A " (/) )  =  (/)
 
Theorem0ima 4779 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
 |-  ( (/) " A )  =  (/)
 
Theoremcsbima12g 4780 Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F " B )  =  ( [_ A  /  x ]_ F "
 [_ A  /  x ]_ B ) )
 
Theoremimadisj 4781 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
 |-  ( ( A " B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
 
Theoremcnvimass 4782 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
 |-  ( `' A " B )  C_  dom  A
 
Theoremcnvimarndm 4783 The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)
 |-  ( `' A " ran  A )  =  dom  A
 
Theoremimasng 4784* The image of a singleton. (Contributed by NM, 8-May-2005.)
 |-  ( A  e.  B  ->  ( R " { A } )  =  {
 y  |  A R y } )
 
Theoremelreimasng 4785 Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
 |-  ( ( Rel  R  /\  A  e.  V ) 
 ->  ( B  e.  ( R " { A }
 ) 
 <->  A R B ) )
 
Theoremelimasn 4786 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( C  e.  ( A " { B }
 ) 
 <-> 
 <. B ,  C >.  e.  A )
 
Theoremelimasng 4787 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
 |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) )
 
Theoremargs 4788* Two ways to express the class of unique-valued arguments of  F, which is the same as the domain of  F whenever  F is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg  F " for this class (for which we have no separate notation). (Contributed by NM, 8-May-2005.)
 |- 
 { x  |  E. y ( F " { x } )  =  { y } }  =  { x  |  E! y  x F y }
 
Theoremeliniseg 4789 Membership in an initial segment. The idiom  ( `' A " { B } ), meaning  { x  |  x A B }, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  C  e.  _V   =>    |-  ( B  e.  V  ->  ( C  e.  ( `' A " { B } )  <->  C A B ) )
 
Theoremepini 4790 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  A  e.  _V   =>    |-  ( `'  _E  " { A } )  =  A
 
Theoreminiseg 4791* An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.)
 |-  ( B  e.  V  ->  ( `' A " { B } )  =  { x  |  x A B } )
 
Theoremdfse2 4792* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
 |-  ( R Se  A  <->  A. x  e.  A  ( A  i^i  ( `' R " { x } ) )  e. 
 _V )
 
Theoremexse2 4793 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( R  e.  V  ->  R Se  A )
 
Theoremimass1 4794 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
 |-  ( A  C_  B  ->  ( A " C )  C_  ( B " C ) )
 
Theoremimass2 4795 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  ( C " A )  C_  ( C " B ) )
 
Theoremndmima 4796 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
 |-  ( -.  A  e.  dom 
 B  ->  ( B " { A } )  =  (/) )
 
Theoremrelcnv 4797 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
 |- 
 Rel  `' A
 
Theoremrelbrcnvg 4798 When  R is a relation, the sethood assumptions on brcnv 4607 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |-  ( Rel  R  ->  ( A `' R B  <->  B R A ) )
 
Theoremrelbrcnv 4799 When  R is a relation, the sethood assumptions on brcnv 4607 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 Rel  R   =>    |-  ( A `' R B 
 <->  B R A )
 
Theoremcotr 4800* Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( R  o.  R )  C_  R  <->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z ) )
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