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Theorem List for Metamath Proof Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsotri 5201 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 5202 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 5203 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 5204 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 5205 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 5206 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 5207 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 5208 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 5209 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 5210 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 5211 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 5212 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 5213 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 5214* 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 5215 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)
 |-  `' (/)  =  (/)
 
Theoremcnvi 5216 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 5217 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 5218 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  `' ( A  \  B )  =  ( `' A  \  `' B )
 
Theoremcnvin 5219 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 5220 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 5221 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 5222 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 5223* 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 5224 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 5225 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
 |-  ( ( A  u.  B ) " C )  =  ( ( A " C )  u.  ( B " C ) )
 
Theoremdminss 5226 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 5227 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 ) ) )
 
Theoreminimass 5228 The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)
 |-  ( ( A  i^i  B ) " C ) 
 C_  ( ( A
 " C )  i^i  ( B " C ) )
 
Theoreminimasn 5229 The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)
 |-  ( C  e.  V  ->  ( ( A  i^i  B ) " { C } )  =  (
 ( A " { C } )  i^i  ( B " { C }
 ) ) )
 
Theoremcnvxp 5230 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 5231 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 5232 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 5233 At least one member of an empty cross product is empty. (Contributed by NM, 27-Aug-2006.)
 |-  ( ( A  X.  B )  =  (/)  <->  ( A  =  (/) 
 \/  B  =  (/) ) )
 
Theoremxpdisj1 5234 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 5235 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 5236 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 5237* 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 5238 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 5239 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 5240 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 5241 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 5242 The range of a square cross product. (Contributed by FL, 17-May-2010.)
 |- 
 ran  ( A  X.  A )  =  A
 
Theoremssxpb 5243 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 5244 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 5245 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)
 |-  ( C  =/=  (/)  ->  (
 ( C  X.  A )  =  ( C  X.  B )  <->  A  =  B ) )
 
Theoremxpcan2 5246 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)
 |-  ( C  =/=  (/)  ->  (
 ( A  X.  C )  =  ( B  X.  C )  <->  A  =  B ) )
 
Theoremxpexr 5247 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 5248 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 5249 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 5250* 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 5251* 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 5252 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 )
 
Theoremxpima 5253 The image by a constant function (or other cross product). (Contributed by Thierry Arnoux, 4-Feb-2017.)
 |-  ( ( A  X.  B ) " C )  =  if (
 ( A  i^i  C )  =  (/) ,  (/) ,  B )
 
Theoremxpima1 5254 The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
 |-  ( ( A  i^i  C )  =  (/)  ->  (
 ( A  X.  B ) " C )  =  (/) )
 
Theoremxpima2 5255 The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
 |-  ( ( A  i^i  C )  =/=  (/)  ->  (
 ( A  X.  B ) " C )  =  B )
 
Theoremxpimasn 5256 The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)
 |-  ( X  e.  A  ->  ( ( A  X.  B ) " { X } )  =  B )
 
Theoremsossfld 5257 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 5258 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 5259 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 5260* 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 5261 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
 |-  ( Rel  R  <->  `' `' R  =  R )
 
Theoremdfrel4v 5262* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5711 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)
 |-  ( Rel  R  <->  R  =  { <. x ,  y >.  |  x R y }
 )
 
Theoremcnvcnv 5263 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 5264 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
 |-  `' `' A  =  ( A  |`  _V )
 
Theoremcnvcnvss 5265 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 5266 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
 |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  `' A  =  `' B ) )
 
Theoremcnveq0 5267 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  `' A  =  (/) ) )
 
Theoremdfrel3 5268 Alternate definition of relation. (Contributed by NM, 14-May-2008.)
 |-  ( Rel  R  <->  ( R  |`  _V )  =  R )
 
Theoremdmresv 5269 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
 |- 
 dom  ( A  |`  _V )  =  dom  A
 
Theoremrnresv 5270 The range of a universal restriction. (Contributed by NM, 14-May-2008.)
 |- 
 ran  ( A  |`  _V )  =  ran  A
 
Theoremdfrn4 5271 Range defined in terms of image. (Contributed by NM, 14-May-2008.)
 |- 
 ran  A  =  ( A " _V )
 
Theoremrescnvcnv 5272 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 5273 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
 |-  `' `' ( A  |`  B )  =  ( `' `' A  |`  B )
 
Theoremimacnvcnv 5274 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
 |-  ( `' `' A " B )  =  ( A " B )
 
Theoremdmsnn0 5275 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 5276 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 5277 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
 |- 
 dom  { (/) }  =  (/)
 
Theoremcnvsn0 5278 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  `' { (/) }  =  (/)
 
Theoremdmsn0el 5279 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 5280 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 5281 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 5282 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 5283 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 5284 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 5285 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 5286 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 5287 Double converse of a singleton of an ordered pair. (Unlike cnvsn 5292, this does not need any sethood assumptions on  A and  B.) (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  `' `' { <. A ,  B >. }  =  `' { <. B ,  A >. }
 
Theoremdmsnsnsn 5288 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 5289 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 5290 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 5291 Extract the first member of an ordered pair. (See op2nda 5294 to extract the second member, op1stb 4698 for an alternate version, and op1st 6294 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 U. dom  { <. A ,  B >. }  =  A
 
Theoremcnvsn 5292 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 5293 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4698 to extract the first member, op2nda 5294 for an alternate version, and op2nd 6295 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 |^| |^| |^| `' { <. A ,  B >. }  =  B
 
Theoremop2nda 5294 Extract the second member of an ordered pair. (See op1sta 5291 to extract the first member, op2ndb 5293 for an alternate version, and op2nd 6295 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 5295 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 5296 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 5297 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5298, elxp6 6317, and elxp7 6318. (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 5298 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5297 when the double intersection does not create class existence problems (caused by int0 4006). (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 5299 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
 |-  ( `' ( F  |`  A ) " B )  =  ( ( `' F " B )  i^i  A )
 
Theoremresdm2 5300 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
 |-  ( A  |`  dom  A )  =  `' `' A
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