HomeHome Intuitionistic Logic Explorer
Theorem List (p. 48 of 106)
< 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 - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelima 4701* 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 4702* 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 4703* 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 4704 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 4705 Deduction version of bound-variable hypothesis builder nfima 4704. (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 4706 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 4707 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 4708 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 )
 
Theoremimai 4709 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
 |-  (  _I  " A )  =  A
 
Theoremrnresi 4710 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 ran  (  _I  |`  A )  =  A
 
Theoremresiima 4711 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
 |-  ( B  C_  A  ->  ( (  _I  |`  A )
 " B )  =  B )
 
Theoremima0 4712 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
 |-  ( A " (/) )  =  (/)
 
Theorem0ima 4713 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
 |-  ( (/) " A )  =  (/)
 
Theoremcsbima12g 4714 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 4715 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 4716 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 4717 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 4718* The image of a singleton. (Contributed by NM, 8-May-2005.)
 |-  ( A  e.  B  ->  ( R " { A } )  =  {
 y  |  A R y } )
 
Theoremelreimasng 4719 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 4720 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 4721 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 4722* 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 4723 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 4724 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 4725* 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 4726* 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 4727 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( R  e.  V  ->  R Se  A )
 
Theoremimass1 4728 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
 |-  ( A  C_  B  ->  ( A " C )  C_  ( B " C ) )
 
Theoremimass2 4729 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 4730 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
 |-  ( -.  A  e.  dom 
 B  ->  ( B " { A } )  =  (/) )
 
Theoremrelcnv 4731 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
 |- 
 Rel  `' A
 
Theoremrelbrcnvg 4732 When  R is a relation, the sethood assumptions on brcnv 4546 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |-  ( Rel  R  ->  ( A `' R B  <->  B R A ) )
 
Theoremrelbrcnv 4733 When  R is a relation, the sethood assumptions on brcnv 4546 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 Rel  R   =>    |-  ( A `' R B 
 <->  B R A )
 
Theoremcotr 4734* 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 4735* 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 4736* 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 4737* 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 4738* Two ways of saying a relation is antisymmetric and reflexive.  U. U. R is the field of a relation by relfld 4874. (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
 ) )
 
Theoremintirr 4739* 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 4740* 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 4741* 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 4742* 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 4743 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 4744 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 4745 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)
 |-  ( R  Po  A  ->  ( R  i^i  (  _I  |`  A ) )  =  (/) )
 
Theoremtrinxp 4746 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 4747 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 4748 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 4749 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 4750 A transitivity relation. (Read 
-. B < A 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 4751 A transitivity relation. (Read A < B and  -. C < B 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 )
 
Theorempoleloe 4752 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 4753 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 ) )
 
Theoremcnvopab 4754* 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 4755 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)
 |-  `' (/)  =  (/)
 
Theoremcnvi 4756 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 4757 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 4758 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  `' ( A  \  B )  =  ( `' A  \  `' B )
 
Theoremcnvin 4759 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 4760 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 4761 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 4762 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 4763* 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 4764 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 4765 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
 |-  ( ( A  u.  B ) " C )  =  ( ( A " C )  u.  ( B " C ) )
 
Theoremdminss 4766 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 4767 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 4768 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 4769 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 4770 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 4771 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 4772* 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 4773* 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 4774 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 4775 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 4776 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 4777 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 4778* 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 4779 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 4780* The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)
 |-  ( E. x  x  e.  A  ->  ran  ( A  X.  B )  =  B )
 
Theoremdmxpss 4781 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 4782 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 4783 The range of a square cross product. (Contributed by FL, 17-May-2010.)
 |- 
 ran  ( A  X.  A )  =  A
 
Theoremssxpbm 4784* 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 4785* 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 4786* 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 4787* 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 4788* 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 4789* 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 4790* 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 4791 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 4792* 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 4793* 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 4794 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 4795 The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
 |-  ( ( A  i^i  C )  =  (/)  ->  (
 ( A  X.  B ) " C )  =  (/) )
 
Theoremxpima2m 4796* 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 4797 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 4798* 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 4799 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
 |-  ( Rel  R  <->  `' `' R  =  R )
 
Theoremdfrel4v 4800* 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 }
 )
    < 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 10501-10511
  Copyright terms: Public domain < Previous  Next >