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Theorem List for Metamath Proof Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremimaundir 5001 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
 |-  ( ( A  u.  B ) " C )  =  ( ( A " C )  u.  ( B " C ) )
 
Theoremdminss 5002 An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.)
 |-  ( dom  R  i^i  A )  C_  ( `' R " ( R " A ) )
 
Theoremimainss 5003 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)
 |-  ( ( R " A )  i^i  B ) 
 C_  ( R "
 ( A  i^i  ( `' R " B ) ) )
 
Theoremcnvxp 5004 The converse of a cross product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  `' ( A  X.  B )  =  ( B  X.  A )
 
Theoremxp0 5005 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
 |-  ( A  X.  (/) )  =  (/)
 
Theoremxpnz 5006 The cross product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.)
 |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
 
Theoremxpeq0 5007 At least one member of an empty cross product is empty. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( A  X.  B )  =  (/)  <->  ( A  =  (/) 
 \/  B  =  (/) ) )
 
Theoremxpdisj1 5008 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
 |-  ( ( A  i^i  B )  =  (/)  ->  (
 ( A  X.  C )  i^i  ( B  X.  D ) )  =  (/) )
 
Theoremxpdisj2 5009 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
 |-  ( ( A  i^i  B )  =  (/)  ->  (
 ( C  X.  A )  i^i  ( D  X.  B ) )  =  (/) )
 
Theoremxpsndisj 5010 Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)
 |-  ( B  =/=  D  ->  ( ( A  X.  { B } )  i^i  ( C  X.  { D } ) )  =  (/) )
 
Theoremdjudisj 5011* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)
 |-  ( ( A  i^i  B )  =  (/)  ->  ( U_ x  e.  A  ( { x }  X.  C )  i^i  U_ y  e.  B  ( { y }  X.  D ) )  =  (/) )
 
Theoremresdisj 5012 A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  i^i  B )  =  (/)  ->  (
 ( C  |`  A )  |`  B )  =  (/) )
 
Theoremrnxp 5013 The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
 |-  ( A  =/=  (/)  ->  ran  (  A  X.  B )  =  B )
 
Theoremdmxpss 5014 The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)
 |- 
 dom  (  A  X.  B )  C_  A
 
Theoremrnxpss 5015 The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 ran  (  A  X.  B )  C_  B
 
Theoremrnxpid 5016 The range of a square cross product. (Contributed by FL, 17-May-2010.)
 |- 
 ran  (  A  X.  A )  =  A
 
Theoremssxpb 5017 A cross-product subclass relationship is equivalent to the relationship for it components. (Contributed by NM, 17-Dec-2008.)
 |-  ( ( A  X.  B )  =/=  (/)  ->  (
 ( A  X.  B )  C_  ( C  X.  D )  <->  ( A  C_  C  /\  B  C_  D ) ) )
 
Theoremxp11 5018 The cross product of non-empty classes is one-to-one. (Contributed by NM, 31-May-2008.)
 |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  ( ( A  X.  B )  =  ( C  X.  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremxpcan 5019 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)
 |-  ( C  =/=  (/)  ->  (
 ( C  X.  A )  =  ( C  X.  B )  <->  A  =  B ) )
 
Theoremxpcan2 5020 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)
 |-  ( C  =/=  (/)  ->  (
 ( A  X.  C )  =  ( B  X.  C )  <->  A  =  B ) )
 
Theoremxpexr 5021 If a cross product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( A  X.  B )  e.  C  ->  ( A  e.  _V  \/  B  e.  _V )
 )
 
Theoremxpexr2 5022 If a nonempty cross product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( ( A  X.  B )  e.  C  /\  ( A  X.  B )  =/=  (/) )  ->  ( A  e.  _V  /\  B  e.  _V ) )
 
Theoremssrnres 5023 Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)
 |-  ( B  C_  ran  (  C  |`  A )  <->  ran  (  C  i^i  ( A  X.  B ) )  =  B )
 
Theoremrninxp 5024* Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ran  (  C  i^i  ( A  X.  B ) )  =  B  <->  A. y  e.  B  E. x  e.  A  x C y )
 
Theoremdminxp 5025* Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)
 |-  ( dom  (  C  i^i  ( A  X.  B ) )  =  A  <->  A. x  e.  A  E. y  e.  B  x C y )
 
Theoremimainrect 5026 Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)
 |-  ( ( G  i^i  ( A  X.  B ) ) " Y )  =  ( ( G
 " ( Y  i^i  A ) )  i^i  B )
 
Theoremsossfld 5027 The base set of a strict order is contained in the field of the relation, except possibly for one element (note that  (/)  Or  { B }). (Contributed by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( A  \  { B } )  C_  ( dom  R  u.  ran  R ) )
 
Theoremsofld 5028 The base set of a nonempty strict order is the same as the field of the relation. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A )  /\  R  =/=  (/) )  ->  A  =  ( dom  R  u.  ran 
 R ) )
 
Theoremsoex 5029 If the relation in a strict order is a set, then the base field is also a set. (Contributed by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( R  Or  A  /\  R  e.  V )  ->  A  e.  _V )
 
Theoremcnvcnv3 5030* The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)
 |-  `' `' R  =  { <. x ,  y >.  |  x R y }
 
Theoremdfrel2 5031 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
 |-  ( Rel  R  <->  `' `' R  =  R )
 
Theoremdfrel4v 5032* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5420 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)
 |-  ( Rel  R  <->  R  =  { <. x ,  y >.  |  x R y }
 )
 
Theoremcnvcnv 5033 The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)
 |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )
 
Theoremcnvcnv2 5034 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
 |-  `' `' A  =  ( A  |`  _V )
 
Theoremcnvcnvss 5035 The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)
 |-  `' `' A  C_  A
 
Theoremcnveqb 5036 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
 |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  `' A  =  `' B ) )
 
Theoremdfrel3 5037 Alternate definition of relation. (Contributed by NM, 14-May-2008.)
 |-  ( Rel  R  <->  ( R  |`  _V )  =  R )
 
Theoremdmresv 5038 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
 |- 
 dom  (  A  |`  _V )  =  dom  A
 
Theoremrnresv 5039 The range of a universal restriction. (Contributed by NM, 14-May-2008.)
 |- 
 ran  (  A  |`  _V )  =  ran  A
 
Theoremdfrn4 5040 Range defined in terms of image. (Contributed by NM, 14-May-2008.)
 |- 
 ran  A  =  ( A " _V )
 
Theoremrescnvcnv 5041 The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( `' `' A  |`  B )  =  ( A  |`  B )
 
Theoremcnvcnvres 5042 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
 |-  `' `' ( A  |`  B )  =  ( `' `' A  |`  B )
 
Theoremimacnvcnv 5043 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
 |-  ( `' `' A " B )  =  ( A " B )
 
Theoremdmsnn0 5044 The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  e.  ( _V  X.  _V )  <->  dom  {  A }  =/= 
 (/) )
 
Theoremrnsnn0 5045 The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.)
 |-  ( A  e.  ( _V  X.  _V )  <->  ran  {  A }  =/= 
 (/) )
 
Theoremdmsn0 5046 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
 |- 
 dom  { (/) }  =  (/)
 
Theoremcnvsn0 5047 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  `' { (/) }  =  (/)
 
Theoremdmsn0el 5048 The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)
 |-  ( (/)  e.  A  ->  dom  {  A }  =  (/) )
 
Theoremrelsn2 5049 A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.)
 |-  A  e.  _V   =>    |-  ( Rel  { A } 
 <-> 
 dom  {  A }  =/= 
 (/) )
 
Theoremdmsnopg 5050 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( B  e.  V  ->  dom  { <. A ,  B >. }  =  { A } )
 
Theoremdmsnopss 5051 The domain of a singleton of an ordered pair is a subset of the singleton of the first member (with no sethood assumptions on  B). (Contributed by Mario Carneiro, 30-Apr-2015.)
 |- 
 dom  { <. A ,  B >. }  C_  { A }
 
Theoremdmpropg 5052 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( B  e.  V  /\  D  e.  W )  ->  dom  { <. A ,  B >. ,  <. C ,  D >. }  =  { A ,  C }
 )
 
Theoremdmsnop 5053 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  B  e.  _V   =>    |-  dom  { <. A ,  B >. }  =  { A }
 
Theoremdmprop 5054 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)
 |-  B  e.  _V   &    |-  D  e.  _V   =>    |- 
 dom  { <. A ,  B >. ,  <. C ,  D >. }  =  { A ,  C }
 
Theoremdmtpop 5055 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)
 |-  B  e.  _V   &    |-  D  e.  _V   &    |-  F  e.  _V   =>    |-  dom  {
 <. A ,  B >. , 
 <. C ,  D >. , 
 <. E ,  F >. }  =  { A ,  C ,  E }
 
Theoremcnvcnvsn 5056 Double converse of a singleton of an ordered pair. (Unlike cnvsn 5061, this does not need any sethood assumptions on  A and  B.) (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  `' `' { <. A ,  B >. }  =  `' { <. B ,  A >. }
 
Theoremdmsnsnsn 5057 The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |- 
 dom  { { { A } } }  =  { A }
 
Theoremrnsnopg 5058 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( A  e.  V  ->  ran  { <. A ,  B >. }  =  { B } )
 
Theoremrnsnop 5059 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   =>    |-  ran  { <. A ,  B >. }  =  { B }
 
Theoremop1sta 5060 Extract the first member of an ordered pair. (See op2nda 5063 to extract the second member, op1stb 4460 for an alternate version, and op1st 5980 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 U. dom  { <. A ,  B >. }  =  A
 
Theoremcnvsn 5061 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  `' { <. A ,  B >. }  =  { <. B ,  A >. }
 
Theoremop2ndb 5062 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4460 to extract the first member, op2nda 5063 for an alternate version, and op2nd 5981 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 |^| |^| |^| `' { <. A ,  B >. }  =  B
 
Theoremop2nda 5063 Extract the second member of an ordered pair. (See op1sta 5060 to extract the first member, op2ndb 5062 for an alternate version, and op2nd 5981 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 U. ran  { <. A ,  B >. }  =  B
 
Theoremcnvsng 5064 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  `' { <. A ,  B >. }  =  { <. B ,  A >. } )
 
Theoremopswap 5065 Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |- 
 U. `' { <. A ,  B >. }  =  <. B ,  A >.
 
Theoremelxp4 5066 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5067, elxp6 6003, and elxp7 6004. (Contributed by NM, 17-Feb-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. U. dom  {  A } ,  U. ran  {  A } >.  /\  ( U. dom  {  A }  e.  B  /\  U. ran  {  A }  e.  C )
 ) )
 
Theoremelxp5 5067 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5066 when the double intersection does not create class existence problems (caused by int0 3774). (Contributed by NM, 1-Aug-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. |^| |^| A ,  U. ran  {  A } >.  /\  ( |^| |^| A  e.  B  /\  U. ran  {  A }  e.  C ) ) )
 
Theoremcnvresima 5068 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
 |-  ( `' ( F  |`  A ) " B )  =  ( ( `' F " B )  i^i  A )
 
Theoremresdm2 5069 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
 |-  ( A  |`  dom  A )  =  `' `' A
 
Theoremresdmres 5070 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
 |-  ( A  |`  dom  (  A  |`  B ) )  =  ( A  |`  B )
 
Theoremimadmres 5071 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
 |-  ( A " dom  (  A  |`  B ) )  =  ( A
 " B )
 
Theoremmptpreima 5072* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( `' F " C )  =  { x  e.  A  |  B  e.  C }
 
Theoremmptiniseg 5073* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( C  e.  V  ->  ( `' F " { C } )  =  { x  e.  A  |  B  =  C } )
 
Theoremdmmpt 5074 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  dom 
 F  =  { x  e.  A  |  B  e.  _V
 }
 
Theoremdmmptss 5075* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  dom 
 F  C_  A
 
Theoremdmmptg 5076* The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
 |-  ( A. x  e.  A  B  e.  V  ->  dom  (  x  e.  A  |->  B )  =  A )
 
Theoremrelco 5077 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
 |- 
 Rel  ( A  o.  B )
 
Theoremdfco2 5078* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
 |-  ( A  o.  B )  =  U_ x  e. 
 _V  ( ( `' B " { x } )  X.  ( A " { x }
 ) )
 
Theoremdfco2a 5079* Generalization of dfco2 5078, where  C can have any value between  dom  A  i^i  ran 
B and  _V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( A  o.  B )  =  U_ x  e.  C  ( ( `' B " { x } )  X.  ( A " { x }
 ) ) )
 
Theoremcoundi 5080 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  o.  ( B  u.  C ) )  =  ( ( A  o.  B )  u.  ( A  o.  C ) )
 
Theoremcoundir 5081 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  u.  B )  o.  C )  =  ( ( A  o.  C )  u.  ( B  o.  C ) )
 
Theoremcores 5082 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ran  B  C_  C  ->  ( ( A  |`  C )  o.  B )  =  ( A  o.  B ) )
 
Theoremresco 5083 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
 |-  ( ( A  o.  B )  |`  C )  =  ( A  o.  ( B  |`  C ) )
 
Theoremimaco 5084 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
 |-  ( ( A  o.  B ) " C )  =  ( A " ( B " C ) )
 
Theoremrnco 5085 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
 |- 
 ran  (  A  o.  B )  =  ran  (  A  |`  ran  B )
 
Theoremrnco2 5086 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
 |- 
 ran  (  A  o.  B )  =  ( A " ran  B )
 
Theoremdmco 5087 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
 |- 
 dom  (  A  o.  B )  =  ( `' B " dom  A )
 
Theoremcoiun 5088* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
 |-  ( A  o.  U_ x  e.  C  B )  =  U_ x  e.  C  ( A  o.  B )
 
Theoremcocnvcnv1 5089 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
 |-  ( `' `' A  o.  B )  =  ( A  o.  B )
 
Theoremcocnvcnv2 5090 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
 |-  ( A  o.  `' `' B )  =  ( A  o.  B )
 
Theoremcores2 5091 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
 |-  ( dom  A  C_  C  ->  ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  B ) )
 
Theoremco02 5092 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
 |-  ( A  o.  (/) )  =  (/)
 
Theoremco01 5093 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
 |-  ( (/)  o.  A )  =  (/)
 
Theoremcoi1 5094 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
 |-  ( Rel  A  ->  ( A  o.  _I  )  =  A )
 
Theoremcoi2 5095 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
 |-  ( Rel  A  ->  (  _I  o.  A )  =  A )
 
Theoremcoires1 5096 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
 |-  ( A  o.  (  _I  |`  B ) )  =  ( A  |`  B )
 
Theoremcoass 5097 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
 |-  ( ( A  o.  B )  o.  C )  =  ( A  o.  ( B  o.  C ) )
 
Theoremrelcnvtr 5098 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  R  ->  ( ( R  o.  R )  C_  R  <->  ( `' R  o.  `' R )  C_  `' R ) )
 
Theoremrelssdmrn 5099 A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
 |-  ( Rel  A  ->  A 
 C_  ( dom  A  X.  ran  A ) )
 
Theoremcnvssrndm 5100 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  `' A  C_  ( ran 
 A  X.  dom  A )
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