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Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuni0c 4001* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
 |-  ( U. A  =  (/)  <->  A. x  e.  A  x  =  (/) )
 
Theoremuni0 4002 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 4298 by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
 |- 
 U. (/)  =  (/)
 
Theoremelssuni 4003 An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.)
 |-  ( A  e.  B  ->  A  C_  U. B )
 
Theoremunissel 4004 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
 |-  ( ( U. A  C_  B  /\  B  e.  A )  ->  U. A  =  B )
 
Theoremunissb 4005* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
 |-  ( U. A  C_  B 
 <-> 
 A. x  e.  A  x  C_  B )
 
Theoremuniss2 4006* A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. See iunss2 4096 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.)
 |-  ( A. x  e.  A  E. y  e.  B  x  C_  y  ->  U. A  C_  U. B )
 
Theoremunidif 4007* If the difference  A  \  B contains the largest members of  A, then the union of the difference is the union of  A. (Contributed by NM, 22-Mar-2004.)
 |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. ( A  \  B )  =  U. A )
 
Theoremssunieq 4008* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
 |-  ( ( A  e.  B  /\  A. x  e.  B  x  C_  A )  ->  A  =  U. B )
 
Theoremunimax 4009* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
 |-  ( A  e.  B  ->  U. { x  e.  B  |  x  C_  A }  =  A )
 
2.1.19  The intersection of a class
 
Syntaxcint 4010 Extend class notation to include the intersection of a class (read: 'intersect  A').
 class  |^| A
 
Definitiondf-int 4011* Define the intersection of a class. Definition 7.35 of [TakeutiZaring] p. 44. For example,  |^| { { 1 ,  3 } ,  { 1 ,  8 } }  =  {
1 }. Compare this with the intersection of two classes, df-in 3287. (Contributed by NM, 18-Aug-1993.)
 |- 
 |^| A  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
 
Theoremdfint2 4012* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
 |- 
 |^| A  =  { x  |  A. y  e.  A  x  e.  y }
 
Theoreminteq 4013 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
 |-  ( A  =  B  -> 
 |^| A  =  |^| B )
 
Theoreminteqi 4014 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
 |-  A  =  B   =>    |-  |^| A  =  |^| B
 
Theoreminteqd 4015 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  |^| A  =  |^| B )
 
Theoremelint 4016* Membership in class intersection. (Contributed by NM, 21-May-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  |^|
 B 
 <-> 
 A. x ( x  e.  B  ->  A  e.  x ) )
 
Theoremelint2 4017* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
 |-  A  e.  _V   =>    |-  ( A  e.  |^|
 B 
 <-> 
 A. x  e.  B  A  e.  x )
 
Theoremelintg 4018* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
 |-  ( A  e.  V  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x )
 )
 
Theoremelinti 4019 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( A  e.  |^| B 
 ->  ( C  e.  B  ->  A  e.  C ) )
 
Theoremnfint 4020 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  F/_ x A   =>    |-  F/_ x |^| A
 
Theoremelintab 4021* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
 |-  A  e.  _V   =>    |-  ( A  e.  |^|
 { x  |  ph }  <->  A. x ( ph  ->  A  e.  x ) )
 
Theoremelintrab 4022* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
 |-  A  e.  _V   =>    |-  ( A  e.  |^|
 { x  e.  B  |  ph }  <->  A. x  e.  B  ( ph  ->  A  e.  x ) )
 
Theoremelintrabg 4023* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
 |-  ( A  e.  V  ->  ( A  e.  |^| { x  e.  B  |  ph
 } 
 <-> 
 A. x  e.  B  ( ph  ->  A  e.  x ) ) )
 
Theoremint0 4024 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
 |- 
 |^| (/)  =  _V
 
Theoremintss1 4025 An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)
 |-  ( A  e.  B  -> 
 |^| B  C_  A )
 
Theoremssint 4026* Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  |^| B  <->  A. x  e.  B  A  C_  x )
 
Theoremssintab 4027* Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( A  C_  |^| { x  |  ph }  <->  A. x ( ph  ->  A  C_  x )
 )
 
Theoremssintub 4028* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
 |-  A  C_  |^| { x  e.  B  |  A  C_  x }
 
Theoremssmin 4029* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
 |-  A  C_  |^| { x  |  ( A  C_  x  /\  ph ) }
 
Theoremintmin 4030* Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( A  e.  B  -> 
 |^| { x  e.  B  |  A  C_  x }  =  A )
 
Theoremintss 4031 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  -> 
 |^| B  C_  |^| A )
 
Theoremintssuni 4032 The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
 |-  ( A  =/=  (/)  ->  |^| A  C_ 
 U. A )
 
Theoremssintrab 4033* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
 |-  ( A  C_  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  ( ph  ->  A  C_  x ) )
 
Theoremunissint 4034 If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4047). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( U. A  C_  |^|
 A 
 <->  ( A  =  (/)  \/ 
 U. A  =  |^| A ) )
 
Theoremintssuni2 4035 Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)
 |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  |^| A  C_  U. B )
 
Theoremintminss 4036* Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  ps )  ->  |^| { x  e.  B  |  ph }  C_  A )
 
Theoremintmin2 4037* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
 |-  A  e.  _V   =>    |-  |^| { x  |  A  C_  x }  =  A
 
Theoremintmin3 4038* Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ps   =>    |-  ( A  e.  V  ->  |^|
 { x  |  ph } 
 C_  A )
 
Theoremintmin4 4039* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
 |-  ( A  C_  |^| { x  |  ph }  ->  |^| { x  |  ( A  C_  x  /\  ph ) }  =  |^|
 { x  |  ph } )
 
Theoremintab 4040* The intersection of a special case of a class abstraction.  y may be free in  ph and  A, which can be thought of a  ph ( y ) and  A ( y ). Typically, abrexex2 5960 or abexssex 5961 can be used to satisfy the second hypothesis. (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
 |-  A  e.  _V   &    |-  { x  |  E. y ( ph  /\  x  =  A ) }  e.  _V   =>    |-  |^| { x  |  A. y ( ph  ->  A  e.  x ) }  =  { x  |  E. y ( ph  /\  x  =  A ) }
 
Theoremint0el 4041 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
 |-  ( (/)  e.  A  -> 
 |^| A  =  (/) )
 
Theoremintun 4042 The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)
 |- 
 |^| ( A  u.  B )  =  ( |^| A  i^i  |^| B )
 
Theoremintpr 4043 The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 |^| { A ,  B }  =  ( A  i^i  B )
 
Theoremintprg 4044 The intersection of a pair is the intersection of its members. Closed form of intpr 4043. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
 
Theoremintsng 4045 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( A  e.  V  -> 
 |^| { A }  =  A )
 
Theoremintsn 4046 The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)
 |-  A  e.  _V   =>    |-  |^| { A }  =  A
 
Theoremuniintsn 4047* Two ways to express " A is a singleton." See also en1 7133, en1b 7134, card1 7811, and eusn 3840. (Contributed by NM, 2-Aug-2010.)
 |-  ( U. A  =  |^|
 A 
 <-> 
 E. x  A  =  { x } )
 
Theoremuniintab 4048 The union and the intersection of a class abstraction are equal exactly when there is a unique satisfying value of  ph ( x ). (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E! x ph  <->  U. { x  |  ph }  =  |^|
 { x  |  ph } )
 
Theoremintunsn 4049 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
 |-  B  e.  _V   =>    |-  |^| ( A  u.  { B } )  =  ( |^| A  i^i  B )
 
Theoremrint0 4050 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
 
Theoremelrint 4051* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  A. y  e.  B  X  e.  y
 ) )
 
Theoremelrint2 4052* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  e.  A  ->  ( X  e.  ( A  i^i  |^| B )  <->  A. y  e.  B  X  e.  y )
 )
 
2.1.20  Indexed union and intersection
 
Syntaxciun 4053 Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation  U. x  e.  A B, with the same union symbol as cuni 3975. While that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses as distinguished symbol  U_ instead of  U. and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
 class  U_ x  e.  A  B
 
Syntaxciin 4054 Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation  |^| x  e.  A B, with the same intersection symbol as cint 4010. Although that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol  |^|_ instead of  |^| and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
 class  |^|_
 x  e.  A  B
 
Definitiondf-iun 4055* Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications,  A is independent of  x (although this is not required by the definition), and  B depends on  x i.e. can be read informally as  B ( x ). We call  x the index,  A the index set, and  B the indexed set. In most books,  x  e.  A is written as a subscript or underneath a union symbol  U.. We use a special union symbol  U_ to make it easier to distinguish from plain class union. In many theorems, you will see that  x and 
A are in the same distinct variable group (meaning  A cannot depend on  x) and that  B and  x do not share a distinct variable group (meaning that can be thought of as  B ( x ) i.e. can be substituted with a class expression containing 
x). An alternate definition tying indexed union to ordinary union is dfiun2 4085. Theorem uniiun 4104 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5953 and funiunfv 5954 are useful when  B is a function. (Contributed by NM, 27-Jun-1998.)
 |-  U_ x  e.  A  B  =  { y  |  E. x  e.  A  y  e.  B }
 
Definitiondf-iin 4056* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4055. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4086. Theorem intiin 4105 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)
 |-  |^|_ x  e.  A  B  =  { y  |  A. x  e.  A  y  e.  B }
 
Theoremeliun 4057* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
 |-  ( A  e.  U_ x  e.  B  C  <->  E. x  e.  B  A  e.  C )
 
Theoremeliin 4058* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
 |-  ( A  e.  V  ->  ( A  e.  |^|_ x  e.  B  C  <->  A. x  e.  B  A  e.  C )
 )
 
Theoremiuncom 4059* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
 |-  U_ x  e.  A  U_ y  e.  B  C  =  U_ y  e.  B  U_ x  e.  A  C
 
Theoremiuncom4 4060 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)
 |-  U_ x  e.  A  U. B  =  U. U_ x  e.  A  B
 
Theoremiunconst 4061* Indexed union of a constant class, i.e. where  B does not depend on  x. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  =/=  (/)  ->  U_ x  e.  A  B  =  B )
 
Theoremiinconst 4062* Indexed intersection of a constant class, i.e. where  B does not depend on  x. (Contributed by Mario Carneiro, 6-Feb-2015.)
 |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  B  =  B )
 
Theoremiuniin 4063* Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  U_ x  e.  A  |^|_
 y  e.  B  C  C_  |^|_ y  e.  B  U_ x  e.  A  C
 
Theoremiunss1 4064* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  C_  B  -> 
 U_ x  e.  A  C  C_  U_ x  e.  B  C )
 
Theoremiinss1 4065* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
 |-  ( A  C_  B  -> 
 |^|_ x  e.  B  C  C_  |^|_ x  e.  A  C )
 
Theoremiuneq1 4066* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
 |-  ( A  =  B  -> 
 U_ x  e.  A  C  =  U_ x  e.  B  C )
 
Theoremiineq1 4067* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
 |-  ( A  =  B  -> 
 |^|_ x  e.  A  C  =  |^|_ x  e.  B  C )
 
Theoremss2iun 4068 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A. x  e.  A  B  C_  C  -> 
 U_ x  e.  A  B  C_  U_ x  e.  A  C )
 
Theoremiuneq2 4069 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
 |-  ( A. x  e.  A  B  =  C  -> 
 U_ x  e.  A  B  =  U_ x  e.  A  C )
 
Theoremiineq2 4070 Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A. x  e.  A  B  =  C  -> 
 |^|_ x  e.  A  B  =  |^|_ x  e.  A  C )
 
Theoremiuneq2i 4071 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
 |-  ( x  e.  A  ->  B  =  C )   =>    |-  U_ x  e.  A  B  =  U_ x  e.  A  C
 
Theoremiineq2i 4072 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
 |-  ( x  e.  A  ->  B  =  C )   =>    |-  |^|_
 x  e.  A  B  =  |^|_ x  e.  A  C
 
Theoremiineq2d 4073 Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  =  C )   =>    |-  ( ph  ->  |^|_ x  e.  A  B  =  |^|_ x  e.  A  C )
 
Theoremiuneq2dv 4074* Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  U_ x  e.  A  B  =  U_ x  e.  A  C )
 
Theoremiineq2dv 4075* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  |^|_ x  e.  A  B  =  |^|_ x  e.  A  C )
 
Theoremiuneq1d 4076* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  C )
 
Theoremiuneq12d 4077* Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
 
Theoremiuneq2d 4078* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  U_ x  e.  A  B  =  U_ x  e.  A  C )
 
Theoremnfiun 4079 Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
 |-  F/_ y A   &    |-  F/_ y B   =>    |-  F/_ y U_ x  e.  A  B
 
Theoremnfiin 4080 Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
 |-  F/_ y A   &    |-  F/_ y B   =>    |-  F/_ y |^|_ x  e.  A  B
 
Theoremnfiu1 4081 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
 |-  F/_ x U_ x  e.  A  B
 
Theoremnfii1 4082 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
 |-  F/_ x |^|_ x  e.  A  B
 
Theoremdfiun2g 4083* Alternate definition of indexed union when  B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A. x  e.  A  B  e.  C  -> 
 U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B }
 )
 
Theoremdfiin2g 4084* Alternate definition of indexed intersection when  B is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
 |-  ( A. x  e.  A  B  e.  C  -> 
 |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B }
 )
 
Theoremdfiun2 4085* Alternate definition of indexed union when  B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  B  e.  _V   =>    |-  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B }
 
Theoremdfiin2 4086* Alternate definition of indexed intersection when  B is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  B  e.  _V   =>    |-  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B }
 
Theoremdfiunv2 4087* Define double indexed union. (Contributed by FL, 6-Nov-2013.)
 |-  U_ x  e.  A  U_ y  e.  B  C  =  { z  |  E. x  e.  A  E. y  e.  B  z  e.  C }
 
Theoremcbviun 4088* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
 |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y  ->  B  =  C )   =>    |-  U_ x  e.  A  B  =  U_ y  e.  A  C
 
Theoremcbviin 4089* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y  ->  B  =  C )   =>    |-  |^|_ x  e.  A  B  =  |^|_ y  e.  A  C
 
Theoremcbviunv 4090* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)
 |-  ( x  =  y 
 ->  B  =  C )   =>    |-  U_ x  e.  A  B  =  U_ y  e.  A  C
 
Theoremcbviinv 4091* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)
 |-  ( x  =  y 
 ->  B  =  C )   =>    |-  |^|_
 x  e.  A  B  =  |^|_ y  e.  A  C
 
Theoremiunss 4092* Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( U_ x  e.  A  B  C_  C  <->  A. x  e.  A  B  C_  C )
 
Theoremssiun 4093* Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( E. x  e.  A  C  C_  B  ->  C  C_  U_ x  e.  A  B )
 
Theoremssiun2 4094 Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( x  e.  A  ->  B  C_  U_ x  e.  A  B )
 
Theoremssiun2s 4095* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
 |-  ( x  =  C  ->  B  =  D )   =>    |-  ( C  e.  A  ->  D  C_  U_ x  e.  A  B )
 
Theoremiunss2 4096* A subclass condition on the members of two indexed classes  C
( x ) and  D ( y ) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 4006. (Contributed by NM, 9-Dec-2004.)
 |-  ( A. x  e.  A  E. y  e.  B  C  C_  D  -> 
 U_ x  e.  A  C  C_  U_ y  e.  B  D )
 
Theoremiunab 4097* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
 |-  U_ x  e.  A  { y  |  ph }  =  { y  |  E. x  e.  A  ph }
 
Theoremiunrab 4098* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
 |-  U_ x  e.  A  { y  e.  B  |  ph }  =  {
 y  e.  B  |  E. x  e.  A  ph
 }
 
Theoremiunxdif2 4099* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
 |-  ( x  =  y 
 ->  C  =  D )   =>    |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) C 
 C_  D  ->  U_ y  e.  ( A  \  B ) D  =  U_ x  e.  A  C )
 
Theoremssiinf 4100 Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x C   =>    |-  ( C  C_  |^|_ x  e.  A  B  <->  A. x  e.  A  C  C_  B )
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