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Theorem List for Metamath Proof Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-ima 4601 Define the image of a class (as restricted by another class). Definition 6.6(2) of [TakeutiZaring] p. 24. For example,  ( F  =  { <. 2 ,  6
>. ,  <. 3 ,  9 >. }  /\  B  =  { 1 ,  2 } )  ->  ( F " B )  =  { 6 } (ex-ima 20642). Contrast with restriction (df-res 4600) and range (df-rn 4599). For an alternate definition, see dfima2 4921. (Contributed by NM, 2-Aug-1994.)
 |-  ( A " B )  =  ran  (  A  |`  B )
 
Definitiondf-fun 4602 Define predicate that determines if some class  A is a function. Definition 10.1 of [Quine] p. 65. For example, the expression  Fun  cos is true once we define cosine (df-cos 12226). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3974 with the maps-to notation (see df-mpt 3976 and df-mpt2 5715). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4603), a function with a given domain and codomain (df-f 4604), a one-to-one function (df-f1 4605), an onto function (df-fo 4606), or a one-to-one onto function (df-f1o 4607). For alternate definitions, see dffun2 5123, dffun3 5124, dffun4 5125, dffun5 5126, dffun6 5128, dffun7 5138, dffun8 5139, and dffun9 5140. (Contributed by NM, 1-Aug-1994.)
 |-  ( Fun  A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  )
 )
 
Definitiondf-fn 4603 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 5247, dffn3 5253, dffn4 5314, and dffn5 5420. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  Fn  B  <->  ( Fun  A  /\  dom  A  =  B ) )
 
Definitiondf-f 4604 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. For alternate definitions, see dff2 5524, dff3 5525, and dff4 5526. (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A --> B 
 <->  ( F  Fn  A  /\  ran  F  C_  B ) )
 
Definitiondf-f1 4605 Define a one-to-one function. For equivalent definitions see dff12 5293 and dff13 5635. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow). (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A -1-1-> B  <-> 
 ( F : A --> B  /\  Fun  `' F ) )
 
Definitiondf-fo 4606 Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). For alternate definitions, see dffo2 5312, dffo3 5527, dffo4 5528, and dffo5 5529. (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A -onto-> B 
 <->  ( F  Fn  A  /\  ran  F  =  B ) )
 
Definitiondf-f1o 4607 Define a one-to-one onto function. For equivalent definitions see dff1o2 5334, dff1o3 5335, dff1o4 5337, and dff1o5 5338. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow). (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
 
Definitiondf-fv 4608* Define the value of a function,  ( F `  A
), also known as function application. For example,  ( cos `  0
)  =  1 (we prove this in cos0 12304 after we define cosine in df-cos 12226). Typically function  F is defined using maps-to notation (see df-mpt 3976 and df-mpt2 5715), but this is not required. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  ( F `  3 )  =  9 (ex-fv 20643). Note that df-ov 5713 will define two-argument functions using ordered pairs as  ( A F B )  =  ( F `  <. A ,  B >. ). Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 4948), our definition apparently does not appear in the literature. However, it is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful (as shown by ndmfv 5405 and fvprc 5374). The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar  F ( A ) notation for a function's value at  A, i.e. " F of  A," but without context-dependent notational ambiguity. Alternate definitions are dffv2 5444 and dffv3 6171. For other alternate definitions (that fail to evaluate to the empty set for proper classes), see fv2 5373, fv3 5393, and fv4 6172. Restricted equivalents that require  F to be a function are shown in funfv 5438 and funfv2 5439. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5418. (Contributed by NM, 1-Aug-1994.)
 |-  ( F `  A )  =  U. { x  |  ( F " { A } )  =  { x } }
 
Definitiondf-isom 4609* Define the isomorphism predicate. We read this as " H is an  R,  S isomorphism of  A onto  B." Normally,  R and  S are ordering relations on  A and  B respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that  R and  S are subscripts. (Contributed by NM, 4-Mar-1997.)
 |-  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  ( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( H `  x ) S ( H `  y ) ) ) )
 
Theoremxpeq1 4610 Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2 4611 Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
 |-  ( A  =  B  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremelxpi 4612* Membership in a cross product. Uses fewer axioms than elxp 4613. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  e.  ( B  X.  C )  ->  E. x E. y ( A  =  <. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C ) ) )
 
Theoremelxp 4613* Membership in a cross product. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  e.  ( B  X.  C )  <->  E. x E. y
 ( A  =  <. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C )
 ) )
 
Theoremelxp2 4614* Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  E. x  e.  B  E. y  e.  C  A  =  <. x ,  y >. )
 
Theoremxpeq12 4615 Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C )  =  ( B  X.  D ) )
 
Theoremxpeq1i 4616 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A  X.  C )  =  ( B  X.  C )
 
Theoremxpeq2i 4617 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C  X.  A )  =  ( C  X.  B )
 
Theoremxpeq12i 4618 Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  X.  C )  =  ( B  X.  D )
 
Theoremxpeq1d 4619 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2d 4620 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremxpeq12d 4621 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 4622 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 4623 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 4624 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 4625 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 4626 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 4627 Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)
 |-  ( A ( C  X.  D ) B  <-> 
 ( A  e.  C  /\  B  e.  D ) )
 
Theoremopelxpi 4628 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 4629 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 4630 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 4631 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 4632* 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 4633 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 4634 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 4635 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 4636 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 4637 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 4638 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 4639* 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 4640* 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 4641 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 4642 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 4643 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 4644 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 4645* 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 4646* 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 4647 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 4648 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 4649 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 4650* 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 4651* 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 4652* Obsolete proof of resiun2 4882 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 4653* 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 4654 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 4655* 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 4656* 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 4657 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
 
Theorembrinxp2 4658 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 4659 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 4660 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 4661 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 4662 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 4663 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 4664 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 4665 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 4666 Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( Rel  R  ->  ( R  Or  { A } 
 <->  -.  A R A ) )
 
Theoremfrsn 4667 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 4668 Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( Rel  R  ->  ( R  We  { A } 
 <->  -.  A R A ) )
 
Theoremopabssxp 4669* 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 4670* The law of concretion for a binary relation. See brab2a 4645 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 4671* 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 4672* 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 4673* 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 4674* 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 4675 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 4676 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 4402. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( On  i^i  ( _V  X.  _V ) )  =  (/)
 
Theoremonnev 4677 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 4678 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremreleqi 4679 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
 |-  A  =  B   =>    |-  ( Rel  A  <->  Rel 
 B )
 
Theoremreleqd 4680 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremnfrel 4681 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 4682 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 4683* 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 4684* 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 4685* 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 4686* 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 4687* 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 4688* 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 4689* 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 4690* 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 4691* 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 4692* 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 4693* A version of eqrelrdv 4690. (Contributed by Rodolfo Medina, 10-Oct-2010.)
 |-  ( ph  ->  ( <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B ) )   =>    |-  ( ( ( Rel 
 A  /\  Rel  B ) 
 /\  ph )  ->  A  =  B )
 
Theoremssrelrel 4694* A subclass relationship determined by ordered triples. Use relrelss 5102 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 4695* Extensionality principle for ordered triples (used by 2-place operations df-oprab 5714), analogous to eqrel 4684. Use relrelss 5102 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 4696* 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 4697 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 4698 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 4699 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 4700 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 )
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