Theorem List for Intuitionistic Logic Explorer - 4601-4700 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | dmcoss 4601 |
Domain of a composition. Theorem 21 of [Suppes]
p. 63. (Contributed by
NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | rncoss 4602 |
Range of a composition. (Contributed by NM, 19-Mar-1998.)
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Theorem | dmcosseq 4603 |
Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof
shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | dmcoeq 4604 |
Domain of a composition. (Contributed by NM, 19-Mar-1998.)
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Theorem | rncoeq 4605 |
Range of a composition. (Contributed by NM, 19-Mar-1998.)
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Theorem | reseq1 4606 |
Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
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Theorem | reseq2 4607 |
Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
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Theorem | reseq1i 4608 |
Equality inference for restrictions. (Contributed by NM,
21-Oct-2014.)
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Theorem | reseq2i 4609 |
Equality inference for restrictions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | reseq12i 4610 |
Equality inference for restrictions. (Contributed by NM,
21-Oct-2014.)
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Theorem | reseq1d 4611 |
Equality deduction for restrictions. (Contributed by NM,
21-Oct-2014.)
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Theorem | reseq2d 4612 |
Equality deduction for restrictions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | reseq12d 4613 |
Equality deduction for restrictions. (Contributed by NM,
21-Oct-2014.)
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Theorem | nfres 4614 |
Bound-variable hypothesis builder for restriction. (Contributed by NM,
15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
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Theorem | csbresg 4615 |
Distribute proper substitution through the restriction of a class.
(Contributed by Alan Sare, 10-Nov-2012.)
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   ![]_ ]_](_urbrack.gif)      ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)    |
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Theorem | res0 4616 |
A restriction to the empty set is empty. (Contributed by NM,
12-Nov-1994.)
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Theorem | opelres 4617 |
Ordered pair membership in a restriction. Exercise 13 of
[TakeutiZaring] p. 25.
(Contributed by NM, 13-Nov-1995.)
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Theorem | brres 4618 |
Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
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Theorem | opelresg 4619 |
Ordered pair membership in a restriction. Exercise 13 of
[TakeutiZaring] p. 25.
(Contributed by NM, 14-Oct-2005.)
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Theorem | brresg 4620 |
Binary relation on a restriction. (Contributed by Mario Carneiro,
4-Nov-2015.)
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Theorem | opres 4621 |
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.)
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Theorem | resieq 4622 |
A restricted identity relation is equivalent to equality in its domain.
(Contributed by NM, 30-Apr-2004.)
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Theorem | opelresi 4623 |
   belongs to a restriction of the identity class iff
belongs to the restricting class. (Contributed by FL, 27-Oct-2008.)
(Revised by NM, 30-Mar-2016.)
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Theorem | resres 4624 |
The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
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Theorem | resundi 4625 |
Distributive law for restriction over union. Theorem 31 of [Suppes]
p. 65. (Contributed by NM, 30-Sep-2002.)
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Theorem | resundir 4626 |
Distributive law for restriction over union. (Contributed by NM,
23-Sep-2004.)
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Theorem | resindi 4627 |
Class restriction distributes over intersection. (Contributed by FL,
6-Oct-2008.)
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Theorem | resindir 4628 |
Class restriction distributes over intersection. (Contributed by NM,
18-Dec-2008.)
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Theorem | inres 4629 |
Move intersection into class restriction. (Contributed by NM,
18-Dec-2008.)
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Theorem | resiun1 4630* |
Distribution of restriction over indexed union. (Contributed by Mario
Carneiro, 29-May-2015.)
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Theorem | resiun2 4631* |
Distribution of restriction over indexed union. (Contributed by Mario
Carneiro, 29-May-2015.)
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Theorem | dmres 4632 |
The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25.
(Contributed by NM, 1-Aug-1994.)
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Theorem | ssdmres 4633 |
A domain restricted to a subclass equals the subclass. (Contributed by
NM, 2-Mar-1997.)
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Theorem | dmresexg 4634 |
The domain of a restriction to a set exists. (Contributed by NM,
7-Apr-1995.)
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Theorem | resss 4635 |
A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25.
(Contributed by NM, 2-Aug-1994.)
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Theorem | rescom 4636 |
Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
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Theorem | ssres 4637 |
Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
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Theorem | ssres2 4638 |
Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | relres 4639 |
A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25.
(Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon,
27-Aug-2011.)
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Theorem | resabs1 4640 |
Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25.
(Contributed by NM, 9-Aug-1994.)
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Theorem | resabs2 4641 |
Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
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Theorem | residm 4642 |
Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
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Theorem | resima 4643 |
A restriction to an image. (Contributed by NM, 29-Sep-2004.)
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Theorem | resima2 4644 |
Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
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Theorem | xpssres 4645 |
Restriction of a constant function (or other cross product). (Contributed
by Stefan O'Rear, 24-Jan-2015.)
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Theorem | elres 4646* |
Membership in a restriction. (Contributed by Scott Fenton,
17-Mar-2011.)
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Theorem | elsnres 4647* |
Memebership in restriction to a singleton. (Contributed by Scott
Fenton, 17-Mar-2011.)
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Theorem | relssres 4648 |
Simplification law for restriction. (Contributed by NM,
16-Aug-1994.)
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Theorem | resdm 4649 |
A relation restricted to its domain equals itself. (Contributed by NM,
12-Dec-2006.)
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Theorem | resexg 4650 |
The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | resex 4651 |
The restriction of a set is a set. (Contributed by Jeff Madsen,
19-Jun-2011.)
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Theorem | resopab 4652* |
Restriction of a class abstraction of ordered pairs. (Contributed by
NM, 5-Nov-2002.)
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Theorem | resiexg 4653 |
The existence of a restricted identity function, proved without using
the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)
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Theorem | iss 4654 |
A subclass of the identity function is the identity function restricted
to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by
Andrew Salmon, 27-Aug-2011.)
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Theorem | resopab2 4655* |
Restriction of a class abstraction of ordered pairs. (Contributed by
NM, 24-Aug-2007.)
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Theorem | resmpt 4656* |
Restriction of the mapping operation. (Contributed by Mario Carneiro,
15-Jul-2013.)
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Theorem | resmpt3 4657* |
Unconditional restriction of the mapping operation. (Contributed by
Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro,
22-Mar-2015.)
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Theorem | dfres2 4658* |
Alternate definition of the restriction operation. (Contributed by
Mario Carneiro, 5-Nov-2013.)
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Theorem | opabresid 4659* |
The restricted identity expressed with the class builder. (Contributed
by FL, 25-Apr-2012.)
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Theorem | mptresid 4660* |
The restricted identity expressed with the "maps to" notation.
(Contributed by FL, 25-Apr-2012.)
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Theorem | dmresi 4661 |
The domain of a restricted identity function. (Contributed by NM,
27-Aug-2004.)
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Theorem | resid 4662 |
Any relation restricted to the universe is itself. (Contributed by NM,
16-Mar-2004.)
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Theorem | imaeq1 4663 |
Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
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Theorem | imaeq2 4664 |
Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
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Theorem | imaeq1i 4665 |
Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
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Theorem | imaeq2i 4666 |
Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
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Theorem | imaeq1d 4667 |
Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
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Theorem | imaeq2d 4668 |
Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
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Theorem | imaeq12d 4669 |
Equality theorem for image. (Contributed by Mario Carneiro,
4-Dec-2016.)
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Theorem | dfima2 4670* |
Alternate definition of image. Compare definition (d) of [Enderton]
p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | dfima3 4671* |
Alternate definition of image. Compare definition (d) of [Enderton]
p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | elimag 4672* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 20-Jan-2007.)
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Theorem | elima 4673* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 19-Apr-2004.)
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Theorem | elima2 4674* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 11-Aug-2004.)
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Theorem | elima3 4675* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 14-Aug-1994.)
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Theorem | nfima 4676 |
Bound-variable hypothesis builder for image. (Contributed by NM,
30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | nfimad 4677 |
Deduction version of bound-variable hypothesis builder nfima 4676.
(Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro,
15-Oct-2016.)
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Theorem | imadmrn 4678 |
The image of the domain of a class is the range of the class.
(Contributed by NM, 14-Aug-1994.)
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Theorem | imassrn 4679 |
The image of a class is a subset of its range. Theorem 3.16(xi) of
[Monk1] p. 39. (Contributed by NM,
31-Mar-1995.)
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Theorem | imaexg 4680 |
The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed
by NM, 24-Jul-1995.)
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Theorem | imai 4681 |
Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38.
(Contributed by NM, 30-Apr-1998.)
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Theorem | rnresi 4682 |
The range of the restricted identity function. (Contributed by NM,
27-Aug-2004.)
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Theorem | resiima 4683 |
The image of a restriction of the identity function. (Contributed by FL,
31-Dec-2006.)
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Theorem | ima0 4684 |
Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed
by NM, 20-May-1998.)
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Theorem | 0ima 4685 |
Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
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Theorem | csbima12g 4686 |
Move class substitution in and out of the image of a function.
(Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro,
4-Dec-2016.)
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   ![]_ ]_](_urbrack.gif)    
   ![]_ ]_](_urbrack.gif)     ![]_ ]_](_urbrack.gif)    |
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Theorem | imadisj 4687 |
A class whose image under another is empty is disjoint with the other's
domain. (Contributed by FL, 24-Jan-2007.)
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Theorem | cnvimass 4688 |
A preimage under any class is included in the domain of the class.
(Contributed by FL, 29-Jan-2007.)
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Theorem | cnvimarndm 4689 |
The preimage of the range of a class is the domain of the class.
(Contributed by Jeff Hankins, 15-Jul-2009.)
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Theorem | imasng 4690* |
The image of a singleton. (Contributed by NM, 8-May-2005.)
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Theorem | elreimasng 4691 |
Elementhood in the image of a singleton. (Contributed by Jim Kingdon,
10-Dec-2018.)
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Theorem | elimasn 4692 |
Membership in an image of a singleton. (Contributed by NM,
15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | elimasng 4693 |
Membership in an image of a singleton. (Contributed by Raph Levien,
21-Oct-2006.)
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Theorem | args 4694* |
Two ways to express the class of unique-valued arguments of ,
which is the same as the domain of whenever is a function.
The left-hand side of the equality is from Definition 10.2 of [Quine]
p. 65. Quine uses the notation "arg " for this class (for which
we have no separate notation). (Contributed by NM, 8-May-2005.)
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Theorem | eliniseg 4695 |
Membership in an initial segment. The idiom        ,
meaning     , is used to specify an initial segment in
(for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by
NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | epini 4696 |
Any set is equal to its preimage under the converse epsilon relation.
(Contributed by Mario Carneiro, 9-Mar-2013.)
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Theorem | iniseg 4697* |
An idiom that signifies an initial segment of an ordering, used, for
example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by
NM, 28-Apr-2004.)
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Theorem | dfse2 4698* |
Alternate definition of set-like relation. (Contributed by Mario
Carneiro, 23-Jun-2015.)
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Theorem | exse2 4699 |
Any set relation is set-like. (Contributed by Mario Carneiro,
22-Jun-2015.)
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Theorem | imass1 4700 |
Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
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