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Theorem List for Metamath Proof Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxpex 4801 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 4802 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 4803 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
 |-  ( Rel  A  ->  Rel  ( A  i^i  B ) )
 
Theoremrelin2 4804 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
 |-  ( Rel  B  ->  Rel  ( A  i^i  B ) )
 
Theoremreldif 4805 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
 |-  ( Rel  A  ->  Rel  ( A  \  B ) )
 
Theoremreliun 4806 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 4807 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 4808* 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 4809* 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 4810 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
 |- 
 Rel  (/)
 
Theoremrelopabi 4811 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)
 |-  A  =  { <. x ,  y >.  |  ph }   =>    |-  Rel 
 A
 
Theoremrelopab 4812 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 4813 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 4814 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |- 
 Rel  _E
 
Theoremopabid2 4815* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
 |-  ( Rel  A  ->  {
 <. x ,  y >.  | 
 <. x ,  y >.  e.  A }  =  A )
 
Theoreminopab 4816* 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 4817* 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 4818 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 4819 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 4820 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 ) )
 
Theoremxpiindi 4821* Distributive law for cross product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( A  =/=  (/)  ->  ( C  X.  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  ( C  X.  B ) )
 
Theoremxpriindi 4822* Distributive law for cross product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( C  X.  ( D  i^i  |^|_ x  e.  A  B ) )  =  ( ( C  X.  D )  i^i  |^|_ x  e.  A  ( C  X.  B ) )
 
Theoremeliunxp 4823* Membership in a union of cross products. Analogue of elxp 4706 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 4824* 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 4825* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4827, 
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 4826* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4828, 
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 4827* 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 4828* 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 4829* 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 4830* Version of ralxp 4827 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 4831* Version of rexxp 4828 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 4832* 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 4833* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2398. (Contributed by NM, 24-Feb-2014.)
 |- 
 Rel  A   &    |-  ( ph  ->  (
 <. x ,  y >.  e.  A  <->  ps ) )   =>    |-  ( ph  ->  A  =  { <. x ,  y >.  |  ps }
 )
 
Theoremrelop 4834* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( Rel  <. A ,  B >. 
 <-> 
 E. x E. y
 ( A  =  { x }  /\  B  =  { x ,  y }
 ) )
 
Theoremideqg 4835 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 4836 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 4837 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 4838 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 4839 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
 |-  ( A  C_  B  ->  ( A  o.  C )  C_  ( B  o.  C ) )
 
Theoremcoss2 4840 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
 |-  ( A  C_  B  ->  ( C  o.  A )  C_  ( C  o.  B ) )
 
Theoremcoeq1 4841 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
 |-  ( A  =  B  ->  ( A  o.  C )  =  ( B  o.  C ) )
 
Theoremcoeq2 4842 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
 |-  ( A  =  B  ->  ( C  o.  A )  =  ( C  o.  B ) )
 
Theoremcoeq1i 4843 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  A  =  B   =>    |-  ( A  o.  C )  =  ( B  o.  C )
 
Theoremcoeq2i 4844 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  A  =  B   =>    |-  ( C  o.  A )  =  ( C  o.  B )
 
Theoremcoeq1d 4845 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  o.  C )  =  ( B  o.  C ) )
 
Theoremcoeq2d 4846 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  o.  A )  =  ( C  o.  B ) )
 
Theoremcoeq12i 4847 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  o.  C )  =  ( B  o.  D )
 
Theoremcoeq12d 4848 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 4849 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  o.  B )
 
Theorembrcog 4850* 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 4851* 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 ) ) )
 
Theorembrco 4852* 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 4853* 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 4854 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  `' A  C_  `' B )
 
Theoremcnveq 4855 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
 |-  ( A  =  B  ->  `' A  =  `' B )
 
Theoremcnveqi 4856 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
 |-  A  =  B   =>    |-  `' A  =  `' B
 
Theoremcnveqd 4857 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  `' A  =  `' B )
 
Theoremelcnv 4858* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
 |-  ( A  e.  `' R 
 <-> 
 E. x E. y
 ( A  =  <. x ,  y >.  /\  y R x ) )
 
Theoremelcnv2 4859* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
 |-  ( A  e.  `' R 
 <-> 
 E. x E. y
 ( A  =  <. x ,  y >.  /\  <. y ,  x >.  e.  R ) )
 
Theoremnfcnv 4860 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x `' A
 
Theoremopelcnvg 4861 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )
 
Theorembrcnvg 4862 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B  <->  B R A ) )
 
Theoremopelcnv 4863 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R )
 
Theorembrcnv 4864 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A `' R B 
 <->  B R A )
 
Theoremcnvco 4865 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
 
Theoremcnvuni 4866* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
 |-  `' U. A  =  U_ x  e.  A  `' x
 
Theoremdfdm3 4867* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
 |- 
 dom  A  =  { x  |  E. y <. x ,  y >.  e.  A }
 
Theoremdfrn2 4868* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
 |- 
 ran  A  =  {
 y  |  E. x  x A y }
 
Theoremdfrn3 4869* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
 |- 
 ran  A  =  {
 y  |  E. x <. x ,  y >.  e.  A }
 
Theoremelrn2g 4870* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B ) )
 
Theoremelrng 4871* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x  x B A ) )
 
Theoremdfdm4 4872 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
 |- 
 dom  A  =  ran  `' A
 
Theoremdfdmf 4873* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   =>    |-  dom  A  =  { x  |  E. y  x A y }
 
Theoremeldmg 4874* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
 
Theoremeldm2g 4875* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B ) )
 
Theoremeldm 4876* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
 |-  A  e.  _V   =>    |-  ( A  e.  dom 
 B 
 <-> 
 E. y  A B y )
 
Theoremeldm2 4877* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  dom 
 B 
 <-> 
 E. y <. A ,  y >.  e.  B )
 
Theoremdmss 4878 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( A  C_  B  ->  dom  A  C_  dom  B )
 
Theoremdmeq 4879 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( A  =  B  ->  dom  A  =  dom  B )
 
Theoremdmeqi 4880 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
 |-  A  =  B   =>    |-  dom  A  =  dom  B
 
Theoremdmeqd 4881 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  dom  A  =  dom  B )
 
Theoremopeldm 4882 Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  C  ->  A  e.  dom  C )
 
Theorembreldm 4883 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A R B  ->  A  e.  dom  R )
 
Theorembreldmg 4884 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
 |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B ) 
 ->  A  e.  dom  R )
 
Theoremdmun 4885 The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 dom  ( A  u.  B )  =  ( dom  A  u.  dom  B )
 
Theoremdmin 4886 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
 |- 
 dom  ( A  i^i  B )  C_  ( dom  A  i^i  dom  B )
 
Theoremdmiun 4887 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |- 
 dom  U_ x  e.  A  B  =  U_ x  e.  A  dom  B
 
Theoremdmuni 4888* The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
 |- 
 dom  U. A  =  U_ x  e.  A  dom  x
 
Theoremdmopab 4889* The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
 |- 
 dom  { <. x ,  y >.  |  ph }  =  { x  |  E. y ph }
 
Theoremdmopabss 4890* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
 |- 
 dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
 
Theoremdmopab3 4891* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
 |-  ( A. x  e.  A  E. y ph  <->  dom  {
 <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
 
Theoremdm0 4892 The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 dom  (/)  =  (/)
 
Theoremdmi 4893 The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 dom  _I  =  _V
 
Theoremdmv 4894 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
 |- 
 dom  _V  =  _V
 
Theoremdm0rn0 4895 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)
 |-  ( dom  A  =  (/)  <->  ran 
 A  =  (/) )
 
Theoremreldm0 4896 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  dom  A  =  (/) ) )
 
Theoremdmxp 4897 The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
 
Theoremdmxpid 4898 The domain of a square cross product. (Contributed by NM, 28-Jul-1995.)
 |- 
 dom  ( A  X.  A )  =  A
 
Theoremdmxpin 4899 The domain of the intersection of two square cross products. Unlike dmin 4886, equality holds. (Contributed by NM, 29-Jan-2008.)
 |- 
 dom  ( ( A  X.  A )  i^i  ( B  X.  B ) )  =  ( A  i^i  B )
 
Theoremxpid11 4900 The cross product of a class with itself is one-to-one. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  X.  A )  =  ( B  X.  B )  <->  A  =  B )
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