HomeHome Intuitionistic Logic Explorer
Theorem List (p. 49 of 114)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnvdif 4801 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  `' ( A  \  B )  =  ( `' A  \  `' B )
 
Theoremcnvin 4802 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 4803 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 4804 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 4805 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 4806* 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 4807 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 4808 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
 |-  ( ( A  u.  B ) " C )  =  ( ( A " C )  u.  ( B " C ) )
 
Theoremdminss 4809 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 4810 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 4811 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 4812 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 4813 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 4814 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.  (/) )  =  (/)
 
Theoremxpmlem 4815* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)
 |-  ( ( E. x  x  e.  A  /\  E. y  y  e.  B ) 
 <-> 
 E. z  z  e.  ( A  X.  B ) )
 
Theoremxpm 4816* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)
 |-  ( ( E. x  x  e.  A  /\  E. y  y  e.  B ) 
 <-> 
 E. z  z  e.  ( A  X.  B ) )
 
Theoremxpeq0r 4817 A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)
 |-  ( ( A  =  (/) 
 \/  B  =  (/) )  ->  ( A  X.  B )  =  (/) )
 
Theoremxpdisj1 4818 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 4819 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 4820 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 4821* 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 4822 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 )  =  (/) )
 
Theoremrnxpm 4823* The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with nonempty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)
 |-  ( E. x  x  e.  A  ->  ran  ( A  X.  B )  =  B )
 
Theoremdmxpss 4824 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 4825 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
 
Theoremdmxpss2 4826 Upper bound for the domain of a binary relation. (Contributed by BJ, 10-Jul-2022.)
 |-  ( R  C_  ( A  X.  B )  ->  dom  R  C_  A )
 
Theoremrnxpss2 4827 Upper bound for the range of a binary relation. (Contributed by BJ, 10-Jul-2022.)
 |-  ( R  C_  ( A  X.  B )  ->  ran  R  C_  B )
 
Theoremrnxpid 4828 The range of a square cross product. (Contributed by FL, 17-May-2010.)
 |- 
 ran  ( A  X.  A )  =  A
 
Theoremssxpbm 4829* A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)
 |-  ( E. x  x  e.  ( A  X.  B )  ->  ( ( A  X.  B ) 
 C_  ( C  X.  D )  <->  ( A  C_  C  /\  B  C_  D ) ) )
 
Theoremssxp1 4830* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
 |-  ( E. x  x  e.  C  ->  (
 ( A  X.  C )  C_  ( B  X.  C )  <->  A  C_  B ) )
 
Theoremssxp2 4831* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
 |-  ( E. x  x  e.  C  ->  (
 ( C  X.  A )  C_  ( C  X.  B )  <->  A  C_  B ) )
 
Theoremxp11m 4832* The cross product of inhabited classes is one-to-one. (Contributed by Jim Kingdon, 13-Dec-2018.)
 |-  ( ( E. x  x  e.  A  /\  E. y  y  e.  B )  ->  ( ( A  X.  B )  =  ( C  X.  D ) 
 <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremxpcanm 4833* Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)
 |-  ( E. x  x  e.  C  ->  (
 ( C  X.  A )  =  ( C  X.  B )  <->  A  =  B ) )
 
Theoremxpcan2m 4834* Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)
 |-  ( E. x  x  e.  C  ->  (
 ( A  X.  C )  =  ( B  X.  C )  <->  A  =  B ) )
 
Theoremxpexr2m 4835* If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.)
 |-  ( ( ( A  X.  B )  e.  C  /\  E. x  x  e.  ( A  X.  B ) )  ->  ( A  e.  _V  /\  B  e.  _V )
 )
 
Theoremssrnres 4836 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 4837* 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 4838* 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 4839 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 )
 
Theoremxpima1 4840 The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
 |-  ( ( A  i^i  C )  =  (/)  ->  (
 ( A  X.  B ) " C )  =  (/) )
 
Theoremxpima2m 4841* The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
 |-  ( E. x  x  e.  ( A  i^i  C )  ->  ( ( A  X.  B ) " C )  =  B )
 
Theoremxpimasn 4842 The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)
 |-  ( X  e.  A  ->  ( ( A  X.  B ) " { X } )  =  B )
 
Theoremcnvcnv3 4843* 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 4844 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
 |-  ( Rel  R  <->  `' `' R  =  R )
 
Theoremdfrel4v 4845* A relation can be expressed as the set of ordered pairs in it. (Contributed by Mario Carneiro, 16-Aug-2015.)
 |-  ( Rel  R  <->  R  =  { <. x ,  y >.  |  x R y }
 )
 
Theoremcnvcnv 4846 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 4847 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
 |-  `' `' A  =  ( A  |`  _V )
 
Theoremcnvcnvss 4848 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 4849 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
 |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  `' A  =  `' B ) )
 
Theoremcnveq0 4850 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  `' A  =  (/) ) )
 
Theoremdfrel3 4851 Alternate definition of relation. (Contributed by NM, 14-May-2008.)
 |-  ( Rel  R  <->  ( R  |`  _V )  =  R )
 
Theoremdmresv 4852 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
 |- 
 dom  ( A  |`  _V )  =  dom  A
 
Theoremrnresv 4853 The range of a universal restriction. (Contributed by NM, 14-May-2008.)
 |- 
 ran  ( A  |`  _V )  =  ran  A
 
Theoremdfrn4 4854 Range defined in terms of image. (Contributed by NM, 14-May-2008.)
 |- 
 ran  A  =  ( A " _V )
 
Theoremcsbrng 4855 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 ran  B  =  ran  [_ A  /  x ]_ B )
 
Theoremrescnvcnv 4856 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 4857 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
 |-  `' `' ( A  |`  B )  =  ( `' `' A  |`  B )
 
Theoremimacnvcnv 4858 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
 |-  ( `' `' A " B )  =  ( A " B )
 
Theoremdmsnm 4859* The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
 |-  ( A  e.  ( _V  X.  _V )  <->  E. x  x  e. 
 dom  { A } )
 
Theoremrnsnm 4860* The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
 |-  ( A  e.  ( _V  X.  _V )  <->  E. x  x  e. 
 ran  { A } )
 
Theoremdmsn0 4861 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
 |- 
 dom  { (/) }  =  (/)
 
Theoremcnvsn0 4862 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  `' { (/) }  =  (/)
 
Theoremdmsn0el 4863 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 }  =  (/) )
 
Theoremrelsn2m 4864* A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)
 |-  A  e.  _V   =>    |-  ( Rel  { A } 
 <-> 
 E. x  x  e. 
 dom  { A } )
 
Theoremdmsnopg 4865 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 } )
 
Theoremdmpropg 4866 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 4867 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 4868 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 4869 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 4870 Double converse of a singleton of an ordered pair. (Unlike cnvsn 4876, this does not need any sethood assumptions on  A and  B.) (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  `' `' { <. A ,  B >. }  =  `' { <. B ,  A >. }
 
Theoremdmsnsnsng 4871 The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)
 |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
 
Theoremrnsnopg 4872 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 } )
 
Theoremrnpropg 4873 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. A ,  C >. ,  <. B ,  D >. }  =  { C ,  D }
 )
 
Theoremrnsnop 4874 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 4875 Extract the first member of an ordered pair. (See op2nda 4878 to extract the second member and op1stb 4272 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 U. dom  { <. A ,  B >. }  =  A
 
Theoremcnvsn 4876 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 4877 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4272 to extract the first member and op2nda 4878 for an alternate version.) (Contributed by NM, 25-Nov-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 |^| |^| |^| `' { <. A ,  B >. }  =  B
 
Theoremop2nda 4878 Extract the second member of an ordered pair. (See op1sta 4875 to extract the first member and op2ndb 4877 for an alternate 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 4879 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  `' { <. A ,  B >. }  =  { <. B ,  A >. } )
 
Theoremopswapg 4880 Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. `' { <. A ,  B >. }  =  <. B ,  A >. )
 
Theoremelxp4 4881 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4882. (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 4882 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4881 when the double intersection does not create class existence problems (caused by int0 3685). (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 4883 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
 |-  ( `' ( F  |`  A ) " B )  =  ( ( `' F " B )  i^i  A )
 
Theoremresdm2 4884 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
 |-  ( A  |`  dom  A )  =  `' `' A
 
Theoremresdmres 4885 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
 |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
 
Theoremimadmres 4886 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
 |-  ( A " dom  ( A  |`  B ) )  =  ( A
 " B )
 
Theoremmptpreima 4887* 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 4888* 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 4889 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 4890* 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 4891* 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 4892 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
 |- 
 Rel  ( A  o.  B )
 
Theoremdfco2 4893* 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 4894* Generalization of dfco2 4893, 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 4895 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 4896 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 4897 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 4898 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
 |-  ( ( A  o.  B )  |`  C )  =  ( A  o.  ( B  |`  C ) )
 
Theoremimaco 4899 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
 |-  ( ( A  o.  B ) " C )  =  ( A " ( B " C ) )
 
Theoremrnco 4900 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
 |- 
 ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11355
  Copyright terms: Public domain < Previous  Next >