P. 38 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3701-3800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elint2 3701*	Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
|- A e. _V   =>    |- ( A e. |^| B <-> A. x e. B A e. x )
Theorem	elintg 3702*	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 ) )
Theorem	elinti 3703	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 ) )
Theorem	nfint 3704	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
Theorem	elintab 3705*	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 ) )
Theorem	elintrab 3706*	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 ) )
Theorem	elintrabg 3707*	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 ) ) )
Theorem	int0 3708	The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
|- |^| (/) = _V
Theorem	intss1 3709	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 )
Theorem	ssint 3710*	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 )
Theorem	ssintab 3711*	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 ) )
Theorem	ssintub 3712*	Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
|- A C_ |^| { x e. B | A C_ x }
Theorem	ssmin 3713*	Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
|- A C_ |^| { x | ( A C_ x /\ ph ) }
Theorem	intmin 3714*	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 )
Theorem	intss 3715	Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
|- ( A C_ B -> |^| B C_ |^| A )
Theorem	intssunim 3716*	The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
|- ( E. x x e. A -> |^| A C_ U. A )
Theorem	ssintrab 3717*	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 ) )
Theorem	intssuni2m 3718*	Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)
|- ( ( A C_ B /\ E. x x e. A ) -> |^| A C_ U. B )
Theorem	intminss 3719*	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 )
Theorem	intmin2 3720*	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
Theorem	intmin3 3721*	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 )
Theorem	intmin4 3722*	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 } )
Theorem	intab 3723*	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 ). (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 ) }
Theorem	int0el 3724	The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
|- ( (/) e. A -> |^| A = (/) )
Theorem	intun 3725	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 )
Theorem	intpr 3726	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 )
Theorem	intprg 3727	The intersection of a pair is the intersection of its members. Closed form of intpr 3726. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
|- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )
Theorem	intsng 3728	Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( A e. V -> |^| { A } = A )
Theorem	intsn 3729	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
Theorem	uniintsnr 3730*	The union and intersection of a singleton are equal. See also eusn 3520. (Contributed by Jim Kingdon, 14-Aug-2018.)
|- ( E. x A = { x } -> U. A = |^| A )
Theorem	uniintabim 3731	The union and the intersection of a class abstraction are equal if there is a unique satisfying value of ph ( x ). (Contributed by Jim Kingdon, 14-Aug-2018.)
|- ( E! x ph -> U. { x | ph } = |^| { x | ph } )
Theorem	intunsn 3732	Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
|- B e. _V   =>    |- |^| ( A u. { B } ) = ( |^| A i^i B )
Theorem	rint0 3733	Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( X = (/) -> ( A i^i |^| X ) = A )
Theorem	elrint 3734*	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 ) )
Theorem	elrint2 3735*	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
Syntax	ciun 3736	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 3659. 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
Syntax	ciin 3737	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 3694. 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
Definition	df-iun 3738*	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 disjoint variable group (meaning A cannot depend on x) and that B and x do not share a disjoint 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 3770. Theorem uniiun 3789 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)
|- U_ x e. A B = { y | E. x e. A y e. B }
Definition	df-iin 3739*	Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3738. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3771. Theorem intiin 3790 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 }
Theorem	eliun 3740*	Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
|- ( A e. U_ x e. B C <-> E. x e. B A e. C )
Theorem	eliin 3741*	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 ) )
Theorem	iuncom 3742*	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
Theorem	iuncom4 3743	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
Theorem	iunconstm 3744*	Indexed union of a constant class, i.e. where B does not depend on x. (Contributed by Jim Kingdon, 15-Aug-2018.)
|- ( E. x x e. A -> U_ x e. A B = B )
Theorem	iinconstm 3745*	Indexed intersection of a constant class, i.e. where B does not depend on x. (Contributed by Jim Kingdon, 19-Dec-2018.)
|- ( E. y y e. A -> |^|_ x e. A B = B )
Theorem	iuniin 3746*	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
Theorem	iunss1 3747*	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 )
Theorem	iinss1 3748*	Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
|- ( A C_ B -> |^|_ x e. B C C_ |^|_ x e. A C )
Theorem	iuneq1 3749*	Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
|- ( A = B -> U_ x e. A C = U_ x e. B C )
Theorem	iineq1 3750*	Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
|- ( A = B -> |^|_ x e. A C = |^|_ x e. B C )
Theorem	ss2iun 3751	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 )
Theorem	iuneq2 3752	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 )
Theorem	iineq2 3753	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 )
Theorem	iuneq2i 3754	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
Theorem	iineq2i 3755	Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
|- ( x e. A -> B = C )   =>    |- |^|_ x e. A B = |^|_ x e. A C
Theorem	iineq2d 3756	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 )
Theorem	iuneq2dv 3757*	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 )
Theorem	iineq2dv 3758*	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 )
Theorem	iuneq1d 3759*	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 )
Theorem	iuneq12d 3760*	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 )
Theorem	iuneq2d 3761*	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 )
Theorem	nfiunxy 3762*	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
Theorem	nfiinxy 3763*	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
Theorem	nfiunya 3764*	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
Theorem	nfiinya 3765*	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
Theorem	nfiu1 3766	Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
|- F/_ x U_ x e. A B
Theorem	nfii1 3767	Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
|- F/_ x |^|_ x e. A B
Theorem	dfiun2g 3768*	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 } )
Theorem	dfiin2g 3769*	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 } )
Theorem	dfiun2 3770*	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 }
Theorem	dfiin2 3771*	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 }
Theorem	dfiunv2 3772*	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 }
Theorem	cbviun 3773*	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
Theorem	cbviin 3774*	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
Theorem	cbviunv 3775*	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
Theorem	cbviinv 3776*	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
Theorem	iunss 3777*	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 )
Theorem	ssiun 3778*	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 )
Theorem	ssiun2 3779	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 )
Theorem	ssiun2s 3780*	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 )
Theorem	iunss2 3781*	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 3690. (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 )
Theorem	iunab 3782*	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 }
Theorem	iunrab 3783*	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 }
Theorem	iunxdif2 3784*	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 )
Theorem	ssiinf 3785	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 )
Theorem	ssiin 3786*	Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
|- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B )
Theorem	iinss 3787*	Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )
Theorem	iinss2 3788	An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
|- ( x e. A -> |^|_ x e. A B C_ B )
Theorem	uniiun 3789*	Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
|- U. A = U_ x e. A x
Theorem	intiin 3790*	Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
|- |^| A = |^|_ x e. A x
Theorem	iunid 3791*	An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
|- U_ x e. A { x } = A
Theorem	iun0 3792	An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- U_ x e. A (/) = (/)
Theorem	0iun 3793	An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- U_ x e. (/) A = (/)
Theorem	0iin 3794	An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
|- |^|_ x e. (/) A = _V
Theorem	viin 3795*	Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
|- |^|_ x e. _V A = { y | A. x y e. A }
Theorem	iunn0m 3796*	There is an inhabited class in an indexed collection B ( x ) iff the indexed union of them is inhabited. (Contributed by Jim Kingdon, 16-Aug-2018.)
|- ( E. x e. A E. y y e. B <-> E. y y e. U_ x e. A B )
Theorem	iinab 3797*	Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
|- |^|_ x e. A { y | ph } = { y | A. x e. A ph }
Theorem	iinrabm 3798*	Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
|- ( E. x x e. A -> |^|_ x e. A { y e. B | ph } = { y e. B | A. x e. A ph } )
Theorem	iunin2 3799*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3789 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
|- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )
Theorem	iunin1 3800*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3789 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )