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Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembrab2ga 4501* The law of concretion for a binary relation. See brab2a 4479 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 4502* 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 4503* 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 4504* 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 4505* 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 ) )
 
Theorem0xp 4506 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 )  =  (/)
 
Theoremcsbxpg 4507 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 ) )
 
Theoremreleq 4508 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremreleqi 4509 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
 |-  A  =  B   =>    |-  ( Rel  A  <->  Rel 
 B )
 
Theoremreleqd 4510 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremnfrel 4511 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
 
Theoremsbcrel 4512 Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. Rel  R  <->  Rel  [_ A  /  x ]_ R ) )
 
Theoremrelss 4513 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 4514* 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 4515* 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 ) ) )
 
Theoremssrel2 4516* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4514 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)
 |-  ( R  C_  ( A  X.  B )  ->  ( R  C_  S  <->  A. x  e.  A  A. y  e.  B  (
 <. x ,  y >.  e.  R  ->  <. x ,  y >.  e.  S ) ) )
 
Theoremrelssi 4517* 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 4518* 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 4519* 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 4520* 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 4521* 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 4522* 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 4523* 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 4524* 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 4525* A version of eqrelrdv 4522. (Contributed by Rodolfo Medina, 10-Oct-2010.)
 |-  ( ph  ->  ( <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B ) )   =>    |-  ( ( ( Rel 
 A  /\  Rel  B ) 
 /\  ph )  ->  A  =  B )
 
Theoremssrelrel 4526* A subclass relationship determined by ordered triples. Use relrelss 4944 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 4527* Extensionality principle for ordered triples, analogous to eqrel 4515. Use relrelss 4944 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 4528* 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 >. )
 
Theoremrelsng 4529 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)
 |-  ( A  e.  V  ->  ( Rel  { A } 
 <->  A  e.  ( _V 
 X.  _V ) ) )
 
Theoremrelsnopg 4530 A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by BJ, 12-Feb-2022.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  Rel  { <. A ,  B >. } )
 
Theoremrelsn 4531 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 4532 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 4533 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 4534 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 4535 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 4536 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( A  C_  B  ->  ( A  X.  C )  C_  ( B  X.  C ) )
 
Theoremxpss2 4537 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( A  C_  B  ->  ( C  X.  A )  C_  ( C  X.  B ) )
 
Theoremxpsspw 4538 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 )
 
Theoremunixpss 4539 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 4540 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 4541 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 4542 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 4543 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
 |-  ( Rel  A  ->  Rel  ( A  i^i  B ) )
 
Theoremrelin2 4544 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
 |-  ( Rel  B  ->  Rel  ( A  i^i  B ) )
 
Theoremreldif 4545 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
 |-  ( Rel  A  ->  Rel  ( A  \  B ) )
 
Theoremreliun 4546 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 4547 An indexed intersection is a relation 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 4548* 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 4549* 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 4550 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
 |- 
 Rel  (/)
 
Theoremrelopabi 4551 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)
 |-  A  =  { <. x ,  y >.  |  ph }   =>    |-  Rel 
 A
 
Theoremrelopab 4552 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 }
 
Theoremreli 4553 The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |- 
 Rel  _I
 
Theoremrele 4554 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |- 
 Rel  _E
 
Theoremopabid2 4555* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
 |-  ( Rel  A  ->  {
 <. x ,  y >.  | 
 <. x ,  y >.  e.  A }  =  A )
 
Theoreminopab 4556* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
 |-  ( { <. x ,  y >.  |  ph }  i^i  {
 <. x ,  y >.  |  ps } )  =  { <. x ,  y >.  |  ( ph  /\  ps ) }
 
Theoremdifopab 4557* The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( { <. x ,  y >.  |  ph }  \  { <. x ,  y >.  |  ps } )  =  { <. x ,  y >.  |  ( ph  /\  -.  ps ) }
 
Theoreminxp 4558 The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  X.  B )  i^i  ( C  X.  D ) )  =  ( ( A  i^i  C )  X.  ( B  i^i  D ) )
 
Theoremxpindi 4559 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
 |-  ( A  X.  ( B  i^i  C ) )  =  ( ( A  X.  B )  i^i  ( A  X.  C ) )
 
Theoremxpindir 4560 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
 |-  ( ( A  i^i  B )  X.  C )  =  ( ( A  X.  C )  i^i  ( B  X.  C ) )
 
Theoremxpiindim 4561* Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
 |-  ( E. y  y  e.  A  ->  ( C  X.  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  ( C  X.  B ) )
 
Theoremxpriindim 4562* Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
 |-  ( E. y  y  e.  A  ->  ( C  X.  ( D  i^i  |^|_
 x  e.  A  B ) )  =  (
 ( C  X.  D )  i^i  |^|_ x  e.  A  ( C  X.  B ) ) )
 
Theoremeliunxp 4563* Membership in a union of cross products. Analogue of elxp 4445 for nonconstant  B ( x ). (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( C  e.  U_ x  e.  A  ( { x }  X.  B ) 
 <-> 
 E. x E. y
 ( C  =  <. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B )
 ) )
 
Theoremopeliunxp2 4564* Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)
 |-  ( x  =  C  ->  B  =  E )   =>    |-  ( <. C ,  D >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) )
 
Theoremraliunxp 4565* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4567, 
B ( y ) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( x  =  <. y ,  z >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  e.  U_  y  e.  A  ( { y }  X.  B ) ph  <->  A. y  e.  A  A. z  e.  B  ps )
 
Theoremrexiunxp 4566* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4568, 
B ( y ) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)
 |-  ( x  =  <. y ,  z >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x  e.  U_  y  e.  A  ( { y }  X.  B ) ph  <->  E. y  e.  A  E. z  e.  B  ps )
 
Theoremralxp 4567* Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)
 |-  ( x  =  <. y ,  z >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. y  e.  A  A. z  e.  B  ps )
 
Theoremrexxp 4568* Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)
 |-  ( x  =  <. y ,  z >.  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x  e.  ( A  X.  B ) ph  <->  E. y  e.  A  E. z  e.  B  ps )
 
Theoremdjussxp 4569* Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
 |-  U_ x  e.  A  ( { x }  X.  B )  C_  ( A  X.  _V )
 
Theoremralxpf 4570* Version of ralxp 4567 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ z ph   &    |-  F/ x ps   &    |-  ( x  = 
 <. y ,  z >.  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  ( A  X.  B ) ph  <->  A. y  e.  A  A. z  e.  B  ps )
 
Theoremrexxpf 4571* Version of rexxp 4568 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ z ph   &    |-  F/ x ps   &    |-  ( x  = 
 <. y ,  z >.  ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  ( A  X.  B ) ph  <->  E. y  e.  A  E. z  e.  B  ps )
 
Theoremiunxpf 4572* Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)
 |-  F/_ y C   &    |-  F/_ z C   &    |-  F/_ x D   &    |-  ( x  =  <. y ,  z >.  ->  C  =  D )   =>    |-  U_ x  e.  ( A  X.  B ) C  =  U_ y  e.  A  U_ z  e.  B  D
 
Theoremopabbi2dv 4573* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2206. (Contributed by NM, 24-Feb-2014.)
 |- 
 Rel  A   &    |-  ( ph  ->  (
 <. x ,  y >.  e.  A  <->  ps ) )   =>    |-  ( ph  ->  A  =  { <. x ,  y >.  |  ps }
 )
 
Theoremrelop 4574* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( Rel  <. A ,  B >. 
 <-> 
 E. x E. y
 ( A  =  { x }  /\  B  =  { x ,  y }
 ) )
 
Theoremideqg 4575 For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
 
Theoremideq 4576 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
 |-  B  e.  _V   =>    |-  ( A  _I  B 
 <->  A  =  B )
 
Theoremididg 4577 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  e.  V  ->  A  _I  A )
 
Theoremissetid 4578 Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  _V  <->  A  _I  A )
 
Theoremcoss1 4579 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
 |-  ( A  C_  B  ->  ( A  o.  C )  C_  ( B  o.  C ) )
 
Theoremcoss2 4580 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
 |-  ( A  C_  B  ->  ( C  o.  A )  C_  ( C  o.  B ) )
 
Theoremcoeq1 4581 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
 |-  ( A  =  B  ->  ( A  o.  C )  =  ( B  o.  C ) )
 
Theoremcoeq2 4582 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
 |-  ( A  =  B  ->  ( C  o.  A )  =  ( C  o.  B ) )
 
Theoremcoeq1i 4583 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  A  =  B   =>    |-  ( A  o.  C )  =  ( B  o.  C )
 
Theoremcoeq2i 4584 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  A  =  B   =>    |-  ( C  o.  A )  =  ( C  o.  B )
 
Theoremcoeq1d 4585 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  o.  C )  =  ( B  o.  C ) )
 
Theoremcoeq2d 4586 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  o.  A )  =  ( C  o.  B ) )
 
Theoremcoeq12i 4587 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  o.  C )  =  ( B  o.  D )
 
Theoremcoeq12d 4588 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  o.  C )  =  ( B  o.  D ) )
 
Theoremnfco 4589 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  o.  B )
 
Theoremelco 4590* Elements of a composed relation. (Contributed by BJ, 10-Jul-2022.)
 |-  ( A  e.  ( R  o.  S )  <->  E. x E. y E. z ( A  =  <. x ,  z >.  /\  ( x S y 
 /\  y R z ) ) )
 
Theorembrcog 4591* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
 
Theoremopelco2g 4592* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  ( C  o.  D )  <->  E. x ( <. A ,  x >.  e.  D  /\  <. x ,  B >.  e.  C ) ) )
 
Theorembrcogw 4593 Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
 |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z )  /\  ( A D X  /\  X C B ) )  ->  A ( C  o.  D ) B )
 
Theoremeqbrrdva 4594* Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
 |-  ( ph  ->  A  C_  ( C  X.  D ) )   &    |-  ( ph  ->  B 
 C_  ( C  X.  D ) )   &    |-  (
 ( ph  /\  x  e.  C  /\  y  e.  D )  ->  ( x A y  <->  x B y ) )   =>    |-  ( ph  ->  A  =  B )
 
Theorembrco 4595* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) )
 
Theoremopelco 4596* Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  ( C  o.  D )  <->  E. x ( A D x  /\  x C B ) )
 
Theoremcnvss 4597 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  `' A  C_  `' B )
 
Theoremcnveq 4598 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
 |-  ( A  =  B  ->  `' A  =  `' B )
 
Theoremcnveqi 4599 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
 |-  A  =  B   =>    |-  `' A  =  `' B
 
Theoremcnveqd 4600 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  `' A  =  `' B )
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