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Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeldm 4601* 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 4602* 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 4603 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( A  C_  B  ->  dom  A  C_  dom  B )
 
Theoremdmeq 4604 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( A  =  B  ->  dom  A  =  dom  B )
 
Theoremdmeqi 4605 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
 |-  A  =  B   =>    |-  dom  A  =  dom  B
 
Theoremdmeqd 4606 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  dom  A  =  dom  B )
 
Theoremopeldm 4607 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 4608 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 4609 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 4610 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 4611 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 4612 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 4613 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 4614* 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 4615* 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 4616* 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 4617* 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 4618 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 4619 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 4620 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
 |- 
 dom  _V  =  _V
 
Theoremdm0rn0 4621 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)
 |-  ( dom  A  =  (/)  <->  ran 
 A  =  (/) )
 
Theoremreldm0 4622 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  dom  A  =  (/) ) )
 
Theoremdmmrnm 4623* 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 4624* 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 4625* The domain of the intersection of two square cross products. Unlike dmin 4612, 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 4626* 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 4627 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 4847). (Contributed by NM, 8-Apr-2007.)
 |- 
 dom  `' `' A  =  dom  A
 
Theoremrncnvcnv 4628 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
 |- 
 ran  `' `' A  =  ran  A
 
Theoremelreldm 4629 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 4630 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
 |-  ( A  =  B  ->  ran  A  =  ran  B )
 
Theoremrneqi 4631 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
 |-  A  =  B   =>    |-  ran  A  =  ran  B
 
Theoremrneqd 4632 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ran  A  =  ran  B )
 
Theoremrnss 4633 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  ran  A  C_  ran  B )
 
Theorembrelrng 4634 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 4635 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 4636 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 4637 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 4638 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 4639 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 4640* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
 |-  ( Rel  R  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
 
Theoremrelelrnb 4641* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
 |-  ( Rel  R  ->  ( A  e.  ran  R  <->  E. x  x R A ) )
 
Theoremreleldmi 4642 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
 |- 
 Rel  R   =>    |-  ( A R B  ->  A  e.  dom  R )
 
Theoremrelelrni 4643 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)
 |- 
 Rel  R   =>    |-  ( A R B  ->  B  e.  ran  R )
 
Theoremdfrnf 4644* Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   =>    |-  ran  A  =  { y  |  E. x  x A y }
 
Theoremelrn2 4645* Membership in a range. (Contributed by NM, 10-Jul-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  ran 
 B 
 <-> 
 E. x <. x ,  A >.  e.  B )
 
Theoremelrn 4646* Membership in a range. (Contributed by NM, 2-Apr-2004.)
 |-  A  e.  _V   =>    |-  ( A  e.  ran 
 B 
 <-> 
 E. x  x B A )
 
Theoremnfdm 4647 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x dom  A
 
Theoremnfrn 4648 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x ran  A
 
Theoremdmiin 4649 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
 |- 
 dom  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  dom  B
 
Theoremrnopab 4650* The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
 |- 
 ran  { <. x ,  y >.  |  ph }  =  { y  |  E. x ph
 }
 
Theoremrnmpt 4651* The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ran 
 F  =  { y  |  E. x  e.  A  y  =  B }
 
Theoremelrnmpt 4652* The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( C  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B )
 )
 
Theoremelrnmpt1s 4653* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( x  =  D  ->  B  =  C )   =>    |-  ( ( D  e.  A  /\  C  e.  V )  ->  C  e.  ran  F )
 
Theoremelrnmpt1 4654 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ( x  e.  A  /\  B  e.  V )  ->  B  e.  ran 
 F )
 
Theoremelrnmptg 4655* Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
 
Theoremelrnmpti 4656* Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  B  e.  _V   =>    |-  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B )
 
Theoremrn0 4657 The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.)
 |- 
 ran  (/)  =  (/)
 
Theoremdfiun3g 4658 Alternate definition of indexed union when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A. x  e.  A  B  e.  C  -> 
 U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
 
Theoremdfiin3g 4659 Alternate definition of indexed intersection when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A. x  e.  A  B  e.  C  -> 
 |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
 
Theoremdfiun3 4660 Alternate definition of indexed union when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  B  e.  _V   =>    |-  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B )
 
Theoremdfiin3 4661 Alternate definition of indexed intersection when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  B  e.  _V   =>    |-  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B )
 
Theoremriinint 4662* Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ( X  e.  V  /\  A. k  e.  I  S  C_  X )  ->  ( X  i^i  |^|_
 k  e.  I  S )  =  |^| ( { X }  u.  ran  (
 k  e.  I  |->  S ) ) )
 
Theoremrelrn0 4663 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  ran  A  =  (/) ) )
 
Theoremdmrnssfld 4664 The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)
 |-  ( dom  A  u.  ran 
 A )  C_  U. U. A
 
Theoremdmexg 4665 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)
 |-  ( A  e.  V  ->  dom  A  e.  _V )
 
Theoremrnexg 4666 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.)
 |-  ( A  e.  V  ->  ran  A  e.  _V )
 
Theoremdmex 4667 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)
 |-  A  e.  _V   =>    |-  dom  A  e.  _V
 
Theoremrnex 4668 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.)
 |-  A  e.  _V   =>    |-  ran  A  e.  _V
 
Theoremiprc 4669 The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set. (Contributed by NM, 1-Jan-2007.)
 |- 
 -.  _I  e.  _V
 
Theoremdmcoss 4670 Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 dom  ( A  o.  B )  C_  dom  B
 
Theoremrncoss 4671 Range of a composition. (Contributed by NM, 19-Mar-1998.)
 |- 
 ran  ( A  o.  B )  C_  ran  A
 
Theoremdmcosseq 4672 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ran  B  C_  dom 
 A  ->  dom  ( A  o.  B )  = 
 dom  B )
 
Theoremdmcoeq 4673 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
 |-  ( dom  A  =  ran  B  ->  dom  ( A  o.  B )  = 
 dom  B )
 
Theoremrncoeq 4674 Range of a composition. (Contributed by NM, 19-Mar-1998.)
 |-  ( dom  A  =  ran  B  ->  ran  ( A  o.  B )  = 
 ran  A )
 
Theoremreseq1 4675 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
 |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
 
Theoremreseq2 4676 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
 |-  ( A  =  B  ->  ( C  |`  A )  =  ( C  |`  B ) )
 
Theoremreseq1i 4677 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  A  =  B   =>    |-  ( A  |`  C )  =  ( B  |`  C )
 
Theoremreseq2i 4678 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  B   =>    |-  ( C  |`  A )  =  ( C  |`  B )
 
Theoremreseq12i 4679 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  |`  C )  =  ( B  |`  D )
 
Theoremreseq1d 4680 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  C ) )
 
Theoremreseq2d 4681 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  |`  A )  =  ( C  |`  B ) )
 
Theoremreseq12d 4682 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )
 
Theoremnfres 4683 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  |`  B )
 
Theoremcsbresg 4684 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B  |`  C )  =  ( [_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) )
 
Theoremres0 4685 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
 |-  ( A  |`  (/) )  =  (/)
 
Theoremopelres 4686 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
 |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  ( C  |`  D )  <->  ( <. A ,  B >.  e.  C  /\  A  e.  D )
 )
 
Theorembrres 4687 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
 |-  B  e.  _V   =>    |-  ( A ( C  |`  D ) B 
 <->  ( A C B  /\  A  e.  D ) )
 
Theoremopelresg 4688 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)
 |-  ( B  e.  V  ->  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
 ( <. A ,  B >.  e.  C  /\  A  e.  D ) ) )
 
Theorembrresg 4689 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
 |-  ( B  e.  V  ->  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) ) )
 
Theoremopres 4690 Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  B  e.  _V   =>    |-  ( A  e.  D  ->  ( <. A ,  B >.  e.  ( C  |`  D )  <->  <. A ,  B >.  e.  C ) )
 
Theoremresieq 4691 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
 |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B (  _I  |`  A ) C 
 <->  B  =  C ) )
 
Theoremopelresi 4692  <. A ,  A >. belongs to a restriction of the identity class iff  A belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)
 |-  ( A  e.  V  ->  ( <. A ,  A >.  e.  (  _I  |`  B )  <->  A  e.  B )
 )
 
Theoremresres 4693 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
 |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
 
Theoremresundi 4694 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
 |-  ( A  |`  ( B  u.  C ) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
 
Theoremresundir 4695 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
 |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
 
Theoremresindi 4696 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
 |-  ( A  |`  ( B  i^i  C ) )  =  ( ( A  |`  B )  i^i  ( A  |`  C ) )
 
Theoremresindir 4697 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
 |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
 
Theoreminres 4698 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
 |-  ( A  i^i  ( B  |`  C ) )  =  ( ( A  i^i  B )  |`  C )
 
Theoremresiun1 4699* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( U_ x  e.  A  B  |`  C )  =  U_ x  e.  A  ( B  |`  C )
 
Theoremresiun2 4700* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( C  |`  U_ x  e.  A  B )  = 
 U_ x  e.  A  ( C  |`  B )
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