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Theorem List for Metamath Proof Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremimai 5001 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
 |-  (  _I  " A )  =  A
 
Theoremrnresi 5002 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 ran  (  _I  |`  A )  =  A
 
Theoremresiima 5003 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
 |-  ( B  C_  A  ->  ( (  _I  |`  A )
 " B )  =  B )
 
Theoremima0 5004 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
 |-  ( A " (/) )  =  (/)
 
Theorem0ima 5005 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
 |-  ( (/) " A )  =  (/)
 
Theoremimadisj 5006 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 5007 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 5008 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 5009* The image of a singleton. (Contributed by NM, 8-May-2005.)
 |-  ( A  e.  B  ->  ( R " { A } )  =  {
 y  |  A R y } )
 
Theoremrelimasn 5010* The image of a singleton. (Contributed by NM, 20-May-1998.)
 |-  ( Rel  R  ->  ( R " { A } )  =  {
 y  |  A R y } )
 
Theoremelrelimasn 5011 Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)
 |-  ( Rel  R  ->  ( B  e.  ( R
 " { A }
 ) 
 <->  A R B ) )
 
Theoremelimasn 5012 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 5013 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 ) )
 
Theoremelimasni 5014 Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)
 |-  ( C  e.  ( A " { B }
 )  ->  B A C )
 
Theoremargs 5015* 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). Observe the resemblance to our df-fv 4675, which was based on the idea in Quine's definition. (Contributed by NM, 8-May-2005.)
 |- 
 { x  |  E. y ( F " { x } )  =  { y } }  =  { x  |  E! y  x F y }
 
Theoremeliniseg 5016 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 5017 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 5018* 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 } )
 
Theoremdffr3 5019* Alternate definition of well-founded relation. Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)
 |-  ( R  Fr  A  <->  A. x ( ( x 
 C_  A  /\  x  =/= 
 (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } )
 )  =  (/) ) )
 
Theoremdfse2 5020* 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 5021 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( R  e.  V  ->  R Se  A )
 
Theoremimass1 5022 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
 |-  ( A  C_  B  ->  ( A " C )  C_  ( B " C ) )
 
Theoremimass2 5023 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 5024 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
 |-  ( -.  A  e.  dom 
 B  ->  ( B " { A } )  =  (/) )
 
Theoremrelcnv 5025 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
 |- 
 Rel  `' A
 
Theoremrelbrcnvg 5026 When  R is a relation, the sethood assumptions on brcnv 4838 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |-  ( Rel  R  ->  ( A `' R B  <->  B R A ) )
 
Theoremeliniseg2 5027 Eliminate the class existence constraint in eliniseg 5016. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.)
 |-  ( Rel  A  ->  ( C  e.  ( `' A " { B } )  <->  C A B ) )
 
Theoremrelbrcnv 5028 When  R is a relation, the sethood assumptions on brcnv 4838 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 Rel  R   =>    |-  ( A `' R B 
 <->  B R A )
 
Theoremcotr 5029* 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 ) )
 
Theoremissref 5030* Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)
 |-  ( (  _I  |`  A ) 
 C_  R  <->  A. x  e.  A  x R x )
 
Theoremcnvsym 5031* Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( `' R  C_  R 
 <-> 
 A. x A. y
 ( x R y 
 ->  y R x ) )
 
Theoremintasym 5032* Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( R  i^i  `' R )  C_  _I  <->  A. x A. y
 ( ( x R y  /\  y R x )  ->  x  =  y ) )
 
Theoremasymref 5033* Two ways of saying a relation is antisymmetric and reflexive.  U. U. R is the field of a relation by relfld 5171. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( R  i^i  `' R )  =  (  _I  |`  U. U. R ) 
 <-> 
 A. x  e.  U. U. R A. y ( ( x R y 
 /\  y R x )  <->  x  =  y
 ) )
 
Theoremasymref2 5034* Two ways of saying a relation is antisymmetric and reflexive. (Contributed by NM, 6-May-2008.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
 |-  ( ( R  i^i  `' R )  =  (  _I  |`  U. U. R ) 
 <->  ( A. x  e. 
 U. U. R x R x  /\  A. x A. y ( ( x R y  /\  y R x )  ->  x  =  y ) ) )
 
Theoremintirr 5035* Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( R  i^i  _I  )  =  (/)  <->  A. x  -.  x R x )
 
Theorembrcodir 5036* Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( `' R  o.  R ) B  <->  E. z ( A R z  /\  B R z ) ) )
 
Theoremcodir 5037* Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)
 |-  ( ( A  X.  B )  C_  ( `' R  o.  R )  <->  A. x  e.  A  A. y  e.  B  E. z ( x R z  /\  y R z ) )
 
Theoremqfto 5038* A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)
 |-  ( ( A  X.  B )  C_  ( R  u.  `' R )  <->  A. x  e.  A  A. y  e.  B  ( x R y  \/  y R x ) )
 
Theoremxpidtr 5039 A square cross product  ( A  X.  A
) is a transitive relation. (Contributed by FL, 31-Jul-2009.)
 |-  ( ( A  X.  A )  o.  ( A  X.  A ) ) 
 C_  ( A  X.  A )
 
Theoremtrin2 5040 The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)
 |-  ( ( ( R  o.  R )  C_  R  /\  ( S  o.  S )  C_  S ) 
 ->  ( ( R  i^i  S )  o.  ( R  i^i  S ) ) 
 C_  ( R  i^i  S ) )
 
Theorempoirr2 5041 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)
 |-  ( R  Po  A  ->  ( R  i^i  (  _I  |`  A ) )  =  (/) )
 
Theoremtrinxp 5042 The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a square cross product is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.)
 |-  ( ( R  o.  R )  C_  R  ->  ( ( R  i^i  ( A  X.  A ) )  o.  ( R  i^i  ( A  X.  A ) ) )  C_  ( R  i^i  ( A  X.  A ) ) )
 
Theoremsoirri 5043 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   =>    |- 
 -.  A R A
 
Theoremsotri 5044 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   =>    |-  ( ( A R B  /\  B R C )  ->  A R C )
 
Theoremson2lpi 5045 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   =>    |- 
 -.  ( A R B  /\  B R A )
 
Theoremsotri2 5046 A transitivity relation. (Read  A  <_  B and  B  < 
C implies  A  <  C.) (Contributed by Mario Carneiro, 10-May-2013.)
 |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   =>    |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  A R C )
 
Theoremsotri3 5047 A transitivity relation. (Read  A  <  B and  B  <_  C implies  A  <  C.) (Contributed by Mario Carneiro, 10-May-2013.)
 |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   =>    |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  A R C )
 
TheoremsoirriOLD 5048 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   =>    |-  -.  A R A
 
TheoremsotriOLD 5049 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A R B  /\  B R C ) 
 ->  A R C )
 
Theoremson2lpiOLD 5050 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   &    |-  B  e.  _V   =>    |- 
 -.  ( A R B  /\  B R A )
 
Theorempoleloe 5051 Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( B  e.  V  ->  ( A ( R  u.  _I  ) B  <-> 
 ( A R B  \/  A  =  B ) ) )
 
Theorempoltletr 5052 Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A R B  /\  B ( R  u.  _I  ) C )  ->  A R C ) )
 
Theoremsomin1 5053 Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X ) )  ->  if ( A R B ,  A ,  B ) ( R  u.  _I  ) A )
 
Theoremsomincom 5054 Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X ) )  ->  if ( A R B ,  A ,  B )  =  if ( B R A ,  B ,  A )
 )
 
Theoremsomin2 5055 Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X ) )  ->  if ( A R B ,  A ,  B ) ( R  u.  _I  ) B )
 
Theoremsoltmin 5056 Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A R if ( B R C ,  B ,  C )  <->  ( A R B  /\  A R C ) ) )
 
Theoremcnvopab 5057* The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  `' { <. x ,  y >.  |  ph }  =  { <. y ,  x >.  |  ph }
 
Theoremcnv0 5058 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)
 |-  `' (/)  =  (/)
 
Theoremcnvi 5059 The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  `'  _I  =  _I
 
Theoremcnvun 5060 The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
 
Theoremcnvdif 5061 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  `' ( A  \  B )  =  ( `' A  \  `' B )
 
Theoremcnvin 5062 Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)
 |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
 
Theoremrnun 5063 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)
 |- 
 ran  (  A  u.  B )  =  ( ran  A  u.  ran  B )
 
Theoremrnin 5064 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
 |- 
 ran  (  A  i^i  B )  C_  ( ran  A  i^i  ran  B )
 
Theoremrniun 5065 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
 |- 
 ran  U_  x  e.  A  B  =  U_ x  e.  A  ran  B
 
Theoremrnuni 5066* The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro, 29-May-2015.)
 |- 
 ran  U.  A  =  U_ x  e.  A  ran  x
 
Theoremimaundi 5067 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
 |-  ( A " ( B  u.  C ) )  =  ( ( A
 " B )  u.  ( A " C ) )
 
Theoremimaundir 5068 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
 |-  ( ( A  u.  B ) " C )  =  ( ( A " C )  u.  ( B " C ) )
 
Theoremdminss 5069 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 5070 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 5071 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 5072 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 5073 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 5074 At least one member of an empty cross product is empty. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( A  X.  B )  =  (/)  <->  ( A  =  (/) 
 \/  B  =  (/) ) )
 
Theoremxpdisj1 5075 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 5076 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 5077 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 5078* 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 5079 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 5080 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 5081 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 5082 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 5083 The range of a square cross product. (Contributed by FL, 17-May-2010.)
 |- 
 ran  (  A  X.  A )  =  A
 
Theoremssxpb 5084 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 5085 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 5086 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)
 |-  ( C  =/=  (/)  ->  (
 ( C  X.  A )  =  ( C  X.  B )  <->  A  =  B ) )
 
Theoremxpcan2 5087 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)
 |-  ( C  =/=  (/)  ->  (
 ( A  X.  C )  =  ( B  X.  C )  <->  A  =  B ) )
 
Theoremxpexr 5088 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 5089 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 5090 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 5091* 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 5092* 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 5093 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 5094 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 5095 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 5096 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 5097* 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 5098 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
 |-  ( Rel  R  <->  `' `' R  =  R )
 
Theoremdfrel4v 5099* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5488 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)
 |-  ( Rel  R  <->  R  =  { <. x ,  y >.  |  x R y }
 )
 
Theoremcnvcnv 5100 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 ) )
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