HomeHome Metamath Proof Explorer
Theorem List (p. 48 of 314)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21458)
  Hilbert Space Explorer  Hilbert Space Explorer
(21459-22981)
  Users' Mathboxes  Users' Mathboxes
(22982-31321)
 

Theorem List for Metamath Proof Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxpeq2d 4701 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremxpeq12d 4702 Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  X.  C )  =  ( B  X.  D ) )
 
Theoremnfxp 4703 Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  X.  B )
 
Theoremcsbxpg 4704 Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  D  -> 
 [_ A  /  x ]_ ( B  X.  C )  =  ( [_ A  /  x ]_ B  X.  [_ A  /  x ]_ C ) )
 
Theorem0nelxp 4705 The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |- 
 -.  (/)  e.  ( A  X.  B )
 
Theorem0nelelxp 4706 A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)
 |-  ( C  e.  ( A  X.  B )  ->  -.  (/)  e.  C )
 
Theoremopelxp 4707 Ordered pair membership in a cross product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( <. A ,  B >.  e.  ( C  X.  D )  <->  ( A  e.  C  /\  B  e.  D ) )
 
Theorembrxp 4708 Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)
 |-  ( A ( C  X.  D ) B  <-> 
 ( A  e.  C  /\  B  e.  D ) )
 
Theoremopelxpi 4709 Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  <. A ,  B >.  e.  ( C  X.  D ) )
 
Theoremopelxp1 4710 The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( <. A ,  B >.  e.  ( C  X.  D )  ->  A  e.  C )
 
Theoremopelxp2 4711 The second member of an ordered pair of classes in a cross product belongs to second cross product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( <. A ,  B >.  e.  ( C  X.  D )  ->  B  e.  D )
 
Theoremotelxp1 4712 The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.)
 |-  ( <. <. A ,  B >. ,  C >.  e.  (
 ( R  X.  S )  X.  T )  ->  A  e.  R )
 
Theoremrabxp 4713* Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)
 |-  ( x  =  <. y ,  z >.  ->  ( ph 
 <->  ps ) )   =>    |-  { x  e.  ( A  X.  B )  |  ph }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  e.  B  /\  ps ) }
 
Theorembrrelex12 4714 A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( Rel  R  /\  A R B ) 
 ->  ( A  e.  _V  /\  B  e.  _V )
 )
 
Theorembrrelex 4715 A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( Rel  R  /\  A R B ) 
 ->  A  e.  _V )
 
Theorembrrelex2 4716 A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( Rel  R  /\  A R B ) 
 ->  B  e.  _V )
 
Theorembrrelexi 4717 The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)
 |- 
 Rel  R   =>    |-  ( A R B  ->  A  e.  _V )
 
Theorembrrelex2i 4718 The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 Rel  R   =>    |-  ( A R B  ->  B  e.  _V )
 
Theoremnprrel 4719 No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
 |- 
 Rel  R   &    |-  -.  A  e.  _V   =>    |-  -.  A R B
 
Theoremfconstmpt 4720* Representation of a constant function using the mapping operation. (Note that  x cannot appear free in  B.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |-  ( A  X.  { B } )  =  ( x  e.  A  |->  B )
 
Theoremvtoclr 4721* Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |- 
 Rel  R   &    |-  ( ( x R y  /\  y R z )  ->  x R z )   =>    |-  ( ( A R B  /\  B R C )  ->  A R C )
 
Theoremopelvvg 4722 Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  <. A ,  B >.  e.  ( _V  X.  _V ) )
 
Theoremopelvv 4723 Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 <. A ,  B >.  e.  ( _V  X.  _V )
 
Theoremopthprc 4724 Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.)
 |-  ( ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } )
 )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } )
 ) 
 <->  ( A  =  C  /\  B  =  D ) )
 
Theorembrel 4725 Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  R  C_  ( C  X.  D )   =>    |-  ( A R B  ->  ( A  e.  C  /\  B  e.  D ) )
 
Theorembrab2a 4726* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D )  /\  ph ) }   =>    |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D ) 
 /\  ps ) )
 
Theoremelxp3 4727* Membership in a cross product. (Contributed by NM, 5-Mar-1995.)
 |-  ( A  e.  ( B  X.  C )  <->  E. x E. y
 ( <. x ,  y >.  =  A  /\  <. x ,  y >.  e.  ( B  X.  C ) ) )
 
Theoremopeliunxp 4728 Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
 |-  ( <. x ,  C >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  C  e.  B ) )
 
Theoremxpundi 4729 Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
 |-  ( A  X.  ( B  u.  C ) )  =  ( ( A  X.  B )  u.  ( A  X.  C ) )
 
Theoremxpundir 4730 Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
 |-  ( ( A  u.  B )  X.  C )  =  ( ( A  X.  C )  u.  ( B  X.  C ) )
 
Theoremxpiundi 4731* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( C  X.  U_ x  e.  A  B )  =  U_ x  e.  A  ( C  X.  B )
 
Theoremxpiundir 4732* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( U_ x  e.  A  B  X.  C )  =  U_ x  e.  A  ( B  X.  C )
 
TheoremresiundiOLD 4733* Obsolete proof of resiun2 4963 as of 5-Apr-2016. Distributive law for cross product over restriction. (Contributed by Mario Carneiro, 11-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( F  |`  U_ x  e.  A  B )  = 
 U_ x  e.  A  ( F  |`  B )
 
Theoremiunxpconst 4734* Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  U_ x  e.  A  ( { x }  X.  B )  =  ( A  X.  B )
 
Theoremxpun 4735 The cross product of two unions. (Contributed by NM, 12-Aug-2004.)
 |-  ( ( A  u.  B )  X.  ( C  u.  D ) )  =  ( ( ( A  X.  C )  u.  ( A  X.  D ) )  u.  ( ( B  X.  C )  u.  ( B  X.  D ) ) )
 
Theoremelvv 4736* Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
 
Theoremelvvv 4737* Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
 |-  ( A  e.  (
 ( _V  X.  _V )  X.  _V )  <->  E. x E. y E. z  A  =  <.
 <. x ,  y >. ,  z >. )
 
Theoremelvvuni 4738 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
 
Theorembrinxp2 4739 Intersection of binary relation with cross product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A ( R  i^i  ( C  X.  D ) ) B  <-> 
 ( A  e.  C  /\  B  e.  D  /\  A R B ) )
 
Theorembrinxp 4740 Intersection of binary relation with cross product. (Contributed by NM, 9-Mar-1997.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B 
 <->  A ( R  i^i  ( C  X.  D ) ) B ) )
 
Theorempoinxp 4741 Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
 |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A ) )  Po  A )
 
Theoremsoinxp 4742 Intersection of total order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
 |-  ( R  Or  A  <->  ( R  i^i  ( A  X.  A ) )  Or  A )
 
Theoremfrinxp 4743 Intersection of well-founded relation with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
 |-  ( R  Fr  A  <->  ( R  i^i  ( A  X.  A ) )  Fr  A )
 
Theoremseinxp 4744 Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( R Se  A  <->  ( R  i^i  ( A  X.  A ) ) Se  A )
 
Theoremweinxp 4745 Intersection of well-ordering with cross product of its field. (Contributed by NM, 9-Mar-1997.) (Revised by Mario Carneiro, 10-Jul-2014.)
 |-  ( R  We  A  <->  ( R  i^i  ( A  X.  A ) )  We  A )
 
Theoremposn 4746 Partial ordering of a singleton. (Contributed by NM, 27-Apr-2009.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( Rel  R  ->  ( R  Po  { A } 
 <->  -.  A R A ) )
 
Theoremsosn 4747 Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( Rel  R  ->  ( R  Or  { A } 
 <->  -.  A R A ) )
 
Theoremfrsn 4748 Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( Rel  R  ->  ( R  Fr  { A } 
 <->  -.  A R A ) )
 
Theoremwesn 4749 Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( Rel  R  ->  ( R  We  { A } 
 <->  -.  A R A ) )
 
Theoremopabssxp 4750* An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)
 |- 
 { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  C_  ( A  X.  B )
 
Theorembrab2ga 4751* The law of concretion for a binary relation. See brab2a 4726 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D )  /\  ph ) }   =>    |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D ) 
 /\  ps ) )
 
Theoremoptocl 4752* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
 |-  D  =  ( B  X.  C )   &    |-  ( <. x ,  y >.  =  A  ->  ( ph  <->  ps ) )   &    |-  ( ( x  e.  B  /\  y  e.  C )  ->  ph )   =>    |-  ( A  e.  D  ->  ps )
 
Theorem2optocl 4753* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
 |-  R  =  ( C  X.  D )   &    |-  ( <. x ,  y >.  =  A  ->  ( ph  <->  ps ) )   &    |-  ( <. z ,  w >.  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 ( ( x  e.  C  /\  y  e.  D )  /\  (
 z  e.  C  /\  w  e.  D )
 )  ->  ph )   =>    |-  ( ( A  e.  R  /\  B  e.  R )  ->  ch )
 
Theorem3optocl 4754* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
 |-  R  =  ( D  X.  F )   &    |-  ( <. x ,  y >.  =  A  ->  ( ph  <->  ps ) )   &    |-  ( <. z ,  w >.  =  B  ->  ( ps  <->  ch ) )   &    |-  ( <. v ,  u >.  =  C  ->  ( ch  <->  th ) )   &    |-  ( ( ( x  e.  D  /\  y  e.  F )  /\  ( z  e.  D  /\  w  e.  F )  /\  ( v  e.  D  /\  u  e.  F ) )  ->  ph )   =>    |-  ( ( A  e.  R  /\  B  e.  R  /\  C  e.  R ) 
 ->  th )
 
Theoremopbrop 4755* Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)
 |-  ( ( ( z  =  A  /\  w  =  B )  /\  (
 v  =  C  /\  u  =  D )
 )  ->  ( ph  <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) ) 
 /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph )
 ) }   =>    |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  ( <. A ,  B >. R <. C ,  D >.  <->  ps ) )
 
Theoremxp0r 4756 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)
 |-  ( (/)  X.  A )  =  (/)
 
Theoremonxpdisj 4757 Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 4483. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( On  i^i  ( _V  X.  _V ) )  =  (/)
 
Theoremonnev 4758 The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.)
 |- 
 On  =/=  _V
 
Theoremreleq 4759 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremreleqi 4760 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
 |-  A  =  B   =>    |-  ( Rel  A  <->  Rel 
 B )
 
Theoremreleqd 4761 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremnfrel 4762 Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x Rel  A
 
Theoremrelss 4763 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)
 |-  ( A  C_  B  ->  ( Rel  B  ->  Rel 
 A ) )
 
Theoremssrel 4764* A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( Rel  A  ->  ( A  C_  B  <->  A. x A. y
 ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B ) ) )
 
Theoremeqrel 4765* Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)
 |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y (
 <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B ) ) )
 
Theoremrelssi 4766* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
 |- 
 Rel  A   &    |-  ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B )   =>    |-  A  C_  B
 
Theoremrelssdv 4767* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremeqrelriv 4768* Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
 |-  ( <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B )   =>    |-  ( ( Rel 
 A  /\  Rel  B ) 
 ->  A  =  B )
 
Theoremeqrelriiv 4769* Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)
 |- 
 Rel  A   &    |-  Rel  B   &    |-  ( <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B )   =>    |-  A  =  B
 
Theoremeqbrriv 4770* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
 |- 
 Rel  A   &    |-  Rel  B   &    |-  ( x A y  <->  x B y )   =>    |-  A  =  B
 
Theoremeqrelrdv 4771* Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)
 |- 
 Rel  A   &    |-  Rel  B   &    |-  ( ph  ->  (
 <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqbrrdv 4772* Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  Rel  B )   &    |-  ( ph  ->  ( x A y  <->  x B y ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqbrrdiv 4773* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)
 |- 
 Rel  A   &    |-  Rel  B   &    |-  ( ph  ->  ( x A y  <->  x B y ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqrelrdv2 4774* A version of eqrelrdv 4771. (Contributed by Rodolfo Medina, 10-Oct-2010.)
 |-  ( ph  ->  ( <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B ) )   =>    |-  ( ( ( Rel 
 A  /\  Rel  B ) 
 /\  ph )  ->  A  =  B )
 
Theoremssrelrel 4775* A subclass relationship determined by ordered triples. Use relrelss 5183 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  C_  (
 ( _V  X.  _V )  X.  _V )  ->  ( A  C_  B  <->  A. x A. y A. z ( <. <. x ,  y >. ,  z >.  e.  A  ->  <. <. x ,  y >. ,  z >.  e.  B ) ) )
 
Theoremeqrelrel 4776* Extensionality principle for ordered triples (used by 2-place operations df-oprab 5796), analogous to eqrel 4765. Use relrelss 5183 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)
 |-  ( ( A  u.  B )  C_  ( ( _V  X.  _V )  X.  _V )  ->  ( A  =  B  <->  A. x A. y A. z ( <. <. x ,  y >. ,  z >.  e.  A  <->  <. <. x ,  y >. ,  z >.  e.  B ) ) )
 
Theoremelrel 4777* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
 |-  ( ( Rel  R  /\  A  e.  R ) 
 ->  E. x E. y  A  =  <. x ,  y >. )
 
Theoremrelsn 4778 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)
 |-  A  e.  _V   =>    |-  ( Rel  { A } 
 <->  A  e.  ( _V 
 X.  _V ) )
 
Theoremrelsnop 4779 A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 Rel  { <. A ,  B >. }
 
Theoremxpss12 4780 Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  C_  B  /\  C  C_  D )  ->  ( A  X.  C )  C_  ( B  X.  D ) )
 
Theoremxpss 4781 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
 |-  ( A  X.  B )  C_  ( _V  X.  _V )
 
Theoremrelxp 4782 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)
 |- 
 Rel  ( A  X.  B )
 
Theoremxpss1 4783 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( A  C_  B  ->  ( A  X.  C )  C_  ( B  X.  C ) )
 
Theoremxpss2 4784 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( A  C_  B  ->  ( C  X.  A )  C_  ( C  X.  B ) )
 
Theoremxpsspw 4785 A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)
 |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
 
TheoremxpsspwOLD 4786 A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
 
Theoremunixpss 4787 The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)
 |- 
 U. U. ( A  X.  B )  C_  ( A  u.  B )
 
Theoremxpexg 4788 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B )  e.  _V )
 
Theoremxpex 4789 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  X.  B )  e.  _V
 
Theoremrelun 4790 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)
 |-  ( Rel  ( A  u.  B )  <->  ( Rel  A  /\  Rel  B ) )
 
Theoremrelin1 4791 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
 |-  ( Rel  A  ->  Rel  ( A  i^i  B ) )
 
Theoremrelin2 4792 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
 |-  ( Rel  B  ->  Rel  ( A  i^i  B ) )
 
Theoremreldif 4793 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
 |-  ( Rel  A  ->  Rel  ( A  \  B ) )
 
Theoremreliun 4794 An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)
 |-  ( Rel  U_ x  e.  A  B  <->  A. x  e.  A  Rel  B )
 
Theoremreliin 4795 An indexed intersection is a relation if if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)
 |-  ( E. x  e.  A  Rel  B  ->  Rel  |^|_ x  e.  A  B )
 
Theoremreluni 4796* The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.)
 |-  ( Rel  U. A  <->  A. x  e.  A  Rel  x )
 
Theoremrelint 4797* The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)
 |-  ( E. x  e.  A  Rel  x  ->  Rel  |^| A )
 
Theoremrel0 4798 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
 |- 
 Rel  (/)
 
Theoremrelopabi 4799 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)
 |-  A  =  { <. x ,  y >.  |  ph }   =>    |-  Rel 
 A
 
Theoremrelopab 4800 A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) (Unnecessary distinct variable restrictions were removed by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)
 |- 
 Rel  { <. x ,  y >.  |  ph }
    < 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 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31321
  Copyright terms: Public domain < Previous  Next >