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Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxpiindim 4501* 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 4502* 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 4503* Membership in a union of cross products. Analogue of elxp 4390 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 4504* 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 4505* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4507, 
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 4506* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4508, 
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 4507* 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 4508* 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 4509* 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 4510* Version of ralxp 4507 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 4511* Version of rexxp 4508 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 4512* 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 4513* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2172. (Contributed by NM, 24-Feb-2014.)
 |- 
 Rel  A   &    |-  ( ph  ->  (
 <. x ,  y >.  e.  A  <->  ps ) )   =>    |-  ( ph  ->  A  =  { <. x ,  y >.  |  ps }
 )
 
Theoremrelop 4514* 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 4515 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 4516 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 4517 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 4518 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 4519 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
 |-  ( A  C_  B  ->  ( A  o.  C )  C_  ( B  o.  C ) )
 
Theoremcoss2 4520 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
 |-  ( A  C_  B  ->  ( C  o.  A )  C_  ( C  o.  B ) )
 
Theoremcoeq1 4521 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
 |-  ( A  =  B  ->  ( A  o.  C )  =  ( B  o.  C ) )
 
Theoremcoeq2 4522 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
 |-  ( A  =  B  ->  ( C  o.  A )  =  ( C  o.  B ) )
 
Theoremcoeq1i 4523 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  A  =  B   =>    |-  ( A  o.  C )  =  ( B  o.  C )
 
Theoremcoeq2i 4524 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  A  =  B   =>    |-  ( C  o.  A )  =  ( C  o.  B )
 
Theoremcoeq1d 4525 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  o.  C )  =  ( B  o.  C ) )
 
Theoremcoeq2d 4526 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  o.  A )  =  ( C  o.  B ) )
 
Theoremcoeq12i 4527 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  o.  C )  =  ( B  o.  D )
 
Theoremcoeq12d 4528 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 4529 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 4530* 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 4531* 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 4532 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 4533* 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 4534* 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 4535* 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 4536 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  `' A  C_  `' B )
 
Theoremcnveq 4537 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
 |-  ( A  =  B  ->  `' A  =  `' B )
 
Theoremcnveqi 4538 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
 |-  A  =  B   =>    |-  `' A  =  `' B
 
Theoremcnveqd 4539 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  `' A  =  `' B )
 
Theoremelcnv 4540* 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 4541* 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 4542 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 4543 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 4544 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 4545 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 4546 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 )
 
Theoremcsbcnvg 4547 Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
 |-  ( A  e.  V  ->  `' [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F )
 
Theoremcnvco 4548 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 4549* 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 4550* 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 4551* 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 4552* 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 4553* 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 4554* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x  x B A ) )
 
Theoremdfdm4 4555 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
 |- 
 dom  A  =  ran  `' A
 
Theoremdfdmf 4556* 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 }
 
Theoremcsbdmg 4557 Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 dom  B  =  dom  [_ A  /  x ]_ B )
 
Theoremeldmg 4558* 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 4559* 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 4560* 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 4561* 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 4562 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( A  C_  B  ->  dom  A  C_  dom  B )
 
Theoremdmeq 4563 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( A  =  B  ->  dom  A  =  dom  B )
 
Theoremdmeqi 4564 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
 |-  A  =  B   =>    |-  dom  A  =  dom  B
 
Theoremdmeqd 4565 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  dom  A  =  dom  B )
 
Theoremopeldm 4566 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 4567 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 )
 
Theoremopeldmg 4568 Membership of first of an ordered pair in a domain. (Contributed by Jim Kingdon, 9-Jul-2019.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  C  ->  A  e.  dom  C )
 )
 
Theorembreldmg 4569 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 4570 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 4571 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 4572 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 4573* 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 4574* 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 4575* 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 4576* 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 4577 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 4578 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 4579 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
 |- 
 dom  _V  =  _V
 
Theoremdm0rn0 4580 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)
 |-  ( dom  A  =  (/)  <->  ran 
 A  =  (/) )
 
Theoremreldm0 4581 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  dom  A  =  (/) ) )
 
Theoremdmmrnm 4582* A domain is inhabited if and only if the range is inhabited. (Contributed by Jim Kingdon, 15-Dec-2018.)
 |-  ( E. x  x  e.  dom  A  <->  E. y  y  e. 
 ran  A )
 
Theoremdmxpm 4583* 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.)
 |-  ( E. x  x  e.  B  ->  dom  ( A  X.  B )  =  A )
 
Theoremdmxpinm 4584* The domain of the intersection of two square cross products. Unlike dmin 4571, equality holds. (Contributed by NM, 29-Jan-2008.)
 |-  ( E. x  x  e.  ( A  i^i  B )  ->  dom  ( ( A  X.  A )  i^i  ( B  X.  B ) )  =  ( A  i^i  B ) )
 
Theoremxpid11m 4585* The cross product of a class with itself is one-to-one. (Contributed by Jim Kingdon, 8-Dec-2018.)
 |-  ( ( E. x  x  e.  A  /\  E. x  x  e.  B )  ->  ( ( A  X.  A )  =  ( B  X.  B ) 
 <->  A  =  B ) )
 
Theoremdmcnvcnv 4586 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 4799). (Contributed by NM, 8-Apr-2007.)
 |- 
 dom  `' `' A  =  dom  A
 
Theoremrncnvcnv 4587 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
 |- 
 ran  `' `' A  =  ran  A
 
Theoremelreldm 4588 The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
 |-  ( ( Rel  A  /\  B  e.  A ) 
 ->  |^| |^| B  e.  dom  A )
 
Theoremrneq 4589 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
 |-  ( A  =  B  ->  ran  A  =  ran  B )
 
Theoremrneqi 4590 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
 |-  A  =  B   =>    |-  ran  A  =  ran  B
 
Theoremrneqd 4591 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ran  A  =  ran  B )
 
Theoremrnss 4592 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  ran  A  C_  ran  B )
 
Theorembrelrng 4593 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
 |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B ) 
 ->  B  e.  ran  C )
 
Theoremopelrng 4594 Membership of second member of an ordered pair in a range. (Contributed by Jim Kingdon, 26-Jan-2019.)
 |-  ( ( A  e.  F  /\  B  e.  G  /\  <. A ,  B >.  e.  C )  ->  B  e.  ran  C )
 
Theorembrelrn 4595 The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A C B  ->  B  e.  ran  C )
 
Theoremopelrn 4596 Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  C  ->  B  e.  ran  C )
 
Theoremreleldm 4597 The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
 |-  ( ( Rel  R  /\  A R B ) 
 ->  A  e.  dom  R )
 
Theoremrelelrn 4598 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
 |-  ( ( Rel  R  /\  A R B ) 
 ->  B  e.  ran  R )
 
Theoremreleldmb 4599* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
 |-  ( Rel  R  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
 
Theoremrelelrnb 4600* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
 |-  ( Rel  R  ->  ( A  e.  ran  R  <->  E. x  x R A ) )
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