Type | Label | Description |
Statement |
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Theorem | csbima12g 5001 |
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 5002 |
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 5003 |
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 5004 |
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 5005* |
The image of a singleton. (Contributed by NM, 8-May-2005.)
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Theorem | elrelimasn 5006 |
Elementhood in the image of a singleton. (Contributed by Mario
Carneiro, 3-Nov-2015.)
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Theorem | elimasn 5007 |
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 5008 |
Membership in an image of a singleton. (Contributed by Raph Levien,
21-Oct-2006.)
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Theorem | args 5009* |
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 5010 |
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 5011 |
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 5012* |
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 5013* |
Alternate definition of set-like relation. (Contributed by Mario
Carneiro, 23-Jun-2015.)
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Theorem | exse2 5014 |
Any set relation is set-like. (Contributed by Mario Carneiro,
22-Jun-2015.)
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Theorem | imass1 5015 |
Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
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Theorem | imass2 5016 |
Subset theorem for image. Exercise 22(a) of [Enderton] p. 53.
(Contributed by NM, 22-Mar-1998.)
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Theorem | ndmima 5017 |
The image of a singleton outside the domain is empty. (Contributed by NM,
22-May-1998.)
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Theorem | relcnv 5018 |
A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed
by NM, 29-Oct-1996.)
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Theorem | relbrcnvg 5019 |
When is a relation,
the sethood assumptions on brcnv 4822 can be
omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
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Theorem | eliniseg2 5020 |
Eliminate the class existence constraint in eliniseg 5010. (Contributed by
Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.)
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Theorem | relbrcnv 5021 |
When is a relation,
the sethood assumptions on brcnv 4822 can be
omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
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Theorem | cotr 5022* |
Two ways of saying a relation is transitive. Definition of transitivity
in [Schechter] p. 51. (Contributed by
NM, 27-Dec-1996.) (Proof
shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | issref 5023* |
Two ways to state a relation is reflexive. Adapted from Tarski.
(Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)
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Theorem | cnvsym 5024* |
Two ways of saying a relation is symmetric. Similar to definition of
symmetry in [Schechter] p. 51.
(Contributed by NM, 28-Dec-1996.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | intasym 5025* |
Two ways of saying a relation is antisymmetric. Definition of
antisymmetry in [Schechter] p. 51.
(Contributed by NM, 9-Sep-2004.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | asymref 5026* |
Two ways of saying a relation is antisymmetric and reflexive.
  is the field of a relation by relfld 5169. (Contributed by
NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | intirr 5027* |
Two ways of saying a relation is irreflexive. Definition of
irreflexivity in [Schechter] p. 51.
(Contributed by NM, 9-Sep-2004.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | brcodir 5028* |
Two ways of saying that two elements have an upper bound. (Contributed
by Mario Carneiro, 3-Nov-2015.)
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Theorem | codir 5029* |
Two ways of saying a relation is directed. (Contributed by Mario
Carneiro, 22-Nov-2013.)
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Theorem | qfto 5030* |
A quantifier-free way of expressing the total order predicate.
(Contributed by Mario Carneiro, 22-Nov-2013.)
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Theorem | xpidtr 5031 |
A square cross product   is a transitive relation.
(Contributed by FL, 31-Jul-2009.)
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Theorem | trin2 5032 |
The intersection of two transitive classes is transitive. (Contributed
by FL, 31-Jul-2009.)
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Theorem | poirr2 5033 |
A partial order relation is irreflexive. (Contributed by Mario
Carneiro, 2-Nov-2015.)
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Theorem | trinxp 5034 |
The relation induced by a transitive relation on a part of its field is
transitive. (Taking the intersection of a relation with a square cross
product is a way to restrict it to a subset of its field.) (Contributed
by FL, 31-Jul-2009.)
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Theorem | soirri 5035 |
A strict order relation is irreflexive. (Contributed by NM,
10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
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Theorem | sotri 5036 |
A strict order relation is a transitive relation. (Contributed by NM,
10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
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Theorem | son2lpi 5037 |
A strict order relation has no 2-cycle loops. (Contributed by NM,
10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
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Theorem | sotri2 5038 |
A transitivity relation. (Read B < A and B < C implies A < C .)
(Contributed by Mario Carneiro, 10-May-2013.)
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Theorem | sotri3 5039 |
A transitivity relation. (Read A < B and C < B implies A < C .)
(Contributed by Mario Carneiro, 10-May-2013.)
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Theorem | poleloe 5040 |
Express "less than or equals" for general strict orders.
(Contributed by
Stefan O'Rear, 17-Jan-2015.)
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Theorem | poltletr 5041 |
Transitive law for general strict orders. (Contributed by Stefan O'Rear,
17-Jan-2015.)
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Theorem | cnvopab 5042* |
The converse of a class abstraction of ordered pairs. (Contributed by
NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | mptcnv 5043* |
The converse of a mapping function. (Contributed by Thierry Arnoux,
16-Jan-2017.)
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Theorem | cnv0 5044 |
The converse of the empty set. (Contributed by NM, 6-Apr-1998.)
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Theorem | cnvi 5045 |
The converse of the identity relation. Theorem 3.7(ii) of [Monk1]
p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | cnvun 5046 |
The converse of a union is the union of converses. Theorem 16 of
[Suppes] p. 62. (Contributed by NM,
25-Mar-1998.) (Proof shortened by
Andrew Salmon, 27-Aug-2011.)
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Theorem | cnvdif 5047 |
Distributive law for converse over set difference. (Contributed by
Mario Carneiro, 26-Jun-2014.)
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Theorem | cnvin 5048 |
Distributive law for converse over intersection. Theorem 15 of [Suppes]
p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro,
26-Jun-2014.)
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Theorem | rnun 5049 |
Distributive law for range over union. Theorem 8 of [Suppes] p. 60.
(Contributed by NM, 24-Mar-1998.)
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Theorem | rnin 5050 |
The range of an intersection belongs the intersection of ranges. Theorem
9 of [Suppes] p. 60. (Contributed by NM,
15-Sep-2004.)
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Theorem | rniun 5051 |
The range of an indexed union. (Contributed by Mario Carneiro,
29-May-2015.)
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Theorem | rnuni 5052* |
The range of a union. Part of Exercise 8 of [Enderton] p. 41.
(Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro,
29-May-2015.)
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Theorem | imaundi 5053 |
Distributive law for image over union. Theorem 35 of [Suppes] p. 65.
(Contributed by NM, 30-Sep-2002.)
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Theorem | imaundir 5054 |
The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
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Theorem | dminss 5055 |
An upper bound for intersection with a domain. Theorem 40 of [Suppes]
p. 66, who calls it "somewhat surprising". (Contributed by
NM,
11-Aug-2004.)
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Theorem | imainss 5056 |
An upper bound for intersection with an image. Theorem 41 of [Suppes]
p. 66. (Contributed by NM, 11-Aug-2004.)
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Theorem | inimass 5057 |
The image of an intersection. (Contributed by Thierry Arnoux,
16-Dec-2017.)
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Theorem | inimasn 5058 |
The intersection of the image of singleton. (Contributed by Thierry
Arnoux, 16-Dec-2017.)
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Theorem | cnvxp 5059 |
The converse of a cross product. Exercise 11 of [Suppes] p. 67.
(Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon,
27-Aug-2011.)
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Theorem | xp0 5060 |
The cross product with the empty set is empty. Part of Theorem 3.13(ii)
of [Monk1] p. 37. (Contributed by NM,
12-Apr-2004.)
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Theorem | xpmlem 5061* |
The cross product of inhabited classes is inhabited. (Contributed by
Jim Kingdon, 11-Dec-2018.)
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Theorem | xpm 5062* |
The cross product of inhabited classes is inhabited. (Contributed by
Jim Kingdon, 13-Dec-2018.)
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Theorem | xpeq0r 5063 |
A cross product is empty if at least one member is empty. (Contributed by
Jim Kingdon, 12-Dec-2018.)
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Theorem | sqxpeq0 5064 |
A Cartesian square is empty iff its member is empty. (Contributed by Jim
Kingdon, 21-Apr-2023.)
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Theorem | xpdisj1 5065 |
Cross products with disjoint sets are disjoint. (Contributed by NM,
13-Sep-2004.)
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Theorem | xpdisj2 5066 |
Cross products with disjoint sets are disjoint. (Contributed by NM,
13-Sep-2004.)
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Theorem | xpsndisj 5067 |
Cross products with two different singletons are disjoint. (Contributed
by NM, 28-Jul-2004.)
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Theorem | djudisj 5068* |
Disjoint unions with disjoint index sets are disjoint. (Contributed by
Stefan O'Rear, 21-Nov-2014.)
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Theorem | resdisj 5069 |
A double restriction to disjoint classes is the empty set. (Contributed
by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | rnxpm 5070* |
The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37,
with nonempty changed to inhabited. (Contributed by Jim Kingdon,
12-Dec-2018.)
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Theorem | dmxpss 5071 |
The domain of a cross product is a subclass of the first factor.
(Contributed by NM, 19-Mar-2007.)
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Theorem | rnxpss 5072 |
The range of a cross product is a subclass of the second factor.
(Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon,
27-Aug-2011.)
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Theorem | dmxpss2 5073 |
Upper bound for the domain of a binary relation. (Contributed by BJ,
10-Jul-2022.)
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Theorem | rnxpss2 5074 |
Upper bound for the range of a binary relation. (Contributed by BJ,
10-Jul-2022.)
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Theorem | rnxpid 5075 |
The range of a square cross product. (Contributed by FL,
17-May-2010.)
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Theorem | ssxpbm 5076* |
A cross-product subclass relationship is equivalent to the relationship
for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)
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Theorem | ssxp1 5077* |
Cross product subset cancellation. (Contributed by Jim Kingdon,
14-Dec-2018.)
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Theorem | ssxp2 5078* |
Cross product subset cancellation. (Contributed by Jim Kingdon,
14-Dec-2018.)
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Theorem | xp11m 5079* |
The cross product of inhabited classes is one-to-one. (Contributed by
Jim Kingdon, 13-Dec-2018.)
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Theorem | xpcanm 5080* |
Cancellation law for cross-product. (Contributed by Jim Kingdon,
14-Dec-2018.)
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Theorem | xpcan2m 5081* |
Cancellation law for cross-product. (Contributed by Jim Kingdon,
14-Dec-2018.)
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Theorem | xpexr2m 5082* |
If a nonempty cross product is a set, so are both of its components.
(Contributed by Jim Kingdon, 14-Dec-2018.)
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Theorem | ssrnres 5083 |
Subset of the range of a restriction. (Contributed by NM,
16-Jan-2006.)
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Theorem | rninxp 5084* |
Range of the intersection with a cross product. (Contributed by NM,
17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | dminxp 5085* |
Domain of the intersection with a cross product. (Contributed by NM,
17-Jan-2006.)
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Theorem | imainrect 5086 |
Image of a relation restricted to a rectangular region. (Contributed by
Stefan O'Rear, 19-Feb-2015.)
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Theorem | xpima1 5087 |
The image by a cross product. (Contributed by Thierry Arnoux,
16-Dec-2017.)
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Theorem | xpima2m 5088* |
The image by a cross product. (Contributed by Thierry Arnoux,
16-Dec-2017.)
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Theorem | xpimasn 5089 |
The image of a singleton by a cross product. (Contributed by Thierry
Arnoux, 14-Jan-2018.)
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Theorem | cnvcnv3 5090* |
The set of all ordered pairs in a class is the same as the double
converse. (Contributed by Mario Carneiro, 16-Aug-2015.)
|
 
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Theorem | dfrel2 5091 |
Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25.
(Contributed by NM, 29-Dec-1996.)
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Theorem | dfrel4v 5092* |
A relation can be expressed as the set of ordered pairs in it.
(Contributed by Mario Carneiro, 16-Aug-2015.)
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Theorem | cnvcnv 5093 |
The double converse of a class strips out all elements that are not
ordered pairs. (Contributed by NM, 8-Dec-2003.)
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Theorem | cnvcnv2 5094 |
The double converse of a class equals its restriction to the universe.
(Contributed by NM, 8-Oct-2007.)
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Theorem | cnvcnvss 5095 |
The double converse of a class is a subclass. Exercise 2 of
[TakeutiZaring] p. 25. (Contributed
by NM, 23-Jul-2004.)
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Theorem | cnveqb 5096 |
Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
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Theorem | cnveq0 5097 |
A relation empty iff its converse is empty. (Contributed by FL,
19-Sep-2011.)
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Theorem | dfrel3 5098 |
Alternate definition of relation. (Contributed by NM, 14-May-2008.)
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Theorem | dmresv 5099 |
The domain of a universal restriction. (Contributed by NM,
14-May-2008.)
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Theorem | rnresv 5100 |
The range of a universal restriction. (Contributed by NM,
14-May-2008.)
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