Theorem List for Intuitionistic Logic Explorer - 4601-4700 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | brinxp2 4601 |
Intersection of binary relation with cross product. (Contributed by NM,
3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
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Theorem | brinxp 4602 |
Intersection of binary relation with cross product. (Contributed by NM,
9-Mar-1997.)
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Theorem | poinxp 4603 |
Intersection of partial order with cross product of its field.
(Contributed by Mario Carneiro, 10-Jul-2014.)
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Theorem | soinxp 4604 |
Intersection of linear order with cross product of its field.
(Contributed by Mario Carneiro, 10-Jul-2014.)
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Theorem | seinxp 4605 |
Intersection of set-like relation with cross product of its field.
(Contributed by Mario Carneiro, 22-Jun-2015.)
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Se
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Theorem | posng 4606 |
Partial ordering of a singleton. (Contributed by Jim Kingdon,
5-Dec-2018.)
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Theorem | sosng 4607 |
Strict linear ordering on a singleton. (Contributed by Jim Kingdon,
5-Dec-2018.)
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Theorem | opabssxp 4608* |
An abstraction relation is a subset of a related cross product.
(Contributed by NM, 16-Jul-1995.)
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Theorem | brab2ga 4609* |
The law of concretion for a binary relation. See brab2a 4587 for alternate
proof. TODO: should one of them be deleted? (Contributed by Mario
Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)
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Theorem | optocl 4610* |
Implicit substitution of class for ordered pair. (Contributed by NM,
5-Mar-1995.)
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Theorem | 2optocl 4611* |
Implicit substitution of classes for ordered pairs. (Contributed by NM,
12-Mar-1995.)
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Theorem | 3optocl 4612* |
Implicit substitution of classes for ordered pairs. (Contributed by NM,
12-Mar-1995.)
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Theorem | opbrop 4613* |
Ordered pair membership in a relation. Special case. (Contributed by
NM, 5-Aug-1995.)
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Theorem | 0xp 4614 |
The cross product with the empty set is empty. Part of Theorem 3.13(ii)
of [Monk1] p. 37. (Contributed by NM,
4-Jul-1994.)
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Theorem | csbxpg 4615 |
Distribute proper substitution through the cross product of two classes.
(Contributed by Alan Sare, 10-Nov-2012.)
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Theorem | releq 4616 |
Equality theorem for the relation predicate. (Contributed by NM,
1-Aug-1994.)
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Theorem | releqi 4617 |
Equality inference for the relation predicate. (Contributed by NM,
8-Dec-2006.)
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Theorem | releqd 4618 |
Equality deduction for the relation predicate. (Contributed by NM,
8-Mar-2014.)
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Theorem | nfrel 4619 |
Bound-variable hypothesis builder for a relation. (Contributed by NM,
31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | sbcrel 4620 |
Distribute proper substitution through a relation predicate. (Contributed
by Alexander van der Vekens, 23-Jul-2017.)
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Theorem | relss 4621 |
Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58.
(Contributed by NM, 15-Aug-1994.)
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Theorem | ssrel 4622* |
A subclass relationship depends only on a relation's ordered pairs.
Theorem 3.2(i) of [Monk1] p. 33.
(Contributed by NM, 2-Aug-1994.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | eqrel 4623* |
Extensionality principle for relations. Theorem 3.2(ii) of [Monk1]
p. 33. (Contributed by NM, 2-Aug-1994.)
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Theorem | ssrel2 4624* |
A subclass relationship depends only on a relation's ordered pairs.
This version of ssrel 4622 is restricted to the relation's domain.
(Contributed by Thierry Arnoux, 25-Jan-2018.)
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Theorem | relssi 4625* |
Inference from subclass principle for relations. (Contributed by NM,
31-Mar-1998.)
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Theorem | relssdv 4626* |
Deduction from subclass principle for relations. (Contributed by NM,
11-Sep-2004.)
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Theorem | eqrelriv 4627* |
Inference from extensionality principle for relations. (Contributed by
FL, 15-Oct-2012.)
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Theorem | eqrelriiv 4628* |
Inference from extensionality principle for relations. (Contributed by
NM, 17-Mar-1995.)
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Theorem | eqbrriv 4629* |
Inference from extensionality principle for relations. (Contributed by
NM, 12-Dec-2006.)
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Theorem | eqrelrdv 4630* |
Deduce equality of relations from equivalence of membership.
(Contributed by Rodolfo Medina, 10-Oct-2010.)
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Theorem | eqbrrdv 4631* |
Deduction from extensionality principle for relations. (Contributed by
Mario Carneiro, 3-Jan-2017.)
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Theorem | eqbrrdiv 4632* |
Deduction from extensionality principle for relations. (Contributed by
Rodolfo Medina, 10-Oct-2010.)
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Theorem | eqrelrdv2 4633* |
A version of eqrelrdv 4630. (Contributed by Rodolfo Medina,
10-Oct-2010.)
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Theorem | ssrelrel 4634* |
A subclass relationship determined by ordered triples. Use relrelss 5060
to express the antecedent in terms of the relation predicate.
(Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon,
27-Aug-2011.)
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Theorem | eqrelrel 4635* |
Extensionality principle for ordered triples, analogous to eqrel 4623.
Use relrelss 5060 to express the antecedent in terms of the
relation
predicate. (Contributed by NM, 17-Dec-2008.)
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Theorem | elrel 4636* |
A member of a relation is an ordered pair. (Contributed by NM,
17-Sep-2006.)
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Theorem | relsng 4637 |
A singleton is a relation iff it is an ordered pair. (Contributed by NM,
24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)
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Theorem | relsnopg 4638 |
A singleton of an ordered pair is a relation. (Contributed by NM,
17-May-1998.) (Revised by BJ, 12-Feb-2022.)
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Theorem | relsn 4639 |
A singleton is a relation iff it is an ordered pair. (Contributed by
NM, 24-Sep-2013.)
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Theorem | relsnop 4640 |
A singleton of an ordered pair is a relation. (Contributed by NM,
17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
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Theorem | xpss12 4641 |
Subset theorem for cross product. Generalization of Theorem 101 of
[Suppes] p. 52. (Contributed by NM,
26-Aug-1995.) (Proof shortened by
Andrew Salmon, 27-Aug-2011.)
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Theorem | xpss 4642 |
A cross product is included in the ordered pair universe. Exercise 3 of
[TakeutiZaring] p. 25. (Contributed
by NM, 2-Aug-1994.)
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Theorem | relxp 4643 |
A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37.
(Contributed by NM, 2-Aug-1994.)
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Theorem | xpss1 4644 |
Subset relation for cross product. (Contributed by Jeff Hankins,
30-Aug-2009.)
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Theorem | xpss2 4645 |
Subset relation for cross product. (Contributed by Jeff Hankins,
30-Aug-2009.)
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Theorem | xpsspw 4646 |
A cross product is included in the power of the power of the union of
its arguments. (Contributed by NM, 13-Sep-2006.)
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Theorem | unixpss 4647 |
The double class union of a cross product is included in the union of its
arguments. (Contributed by NM, 16-Sep-2006.)
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Theorem | xpexg 4648 |
The cross product of two sets is a set. Proposition 6.2 of
[TakeutiZaring] p. 23. (Contributed
by NM, 14-Aug-1994.)
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Theorem | xpex 4649 |
The cross product of two sets is a set. Proposition 6.2 of
[TakeutiZaring] p. 23.
(Contributed by NM, 14-Aug-1994.)
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Theorem | sqxpexg 4650 |
The Cartesian square of a set is a set. (Contributed by AV,
13-Jan-2020.)
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Theorem | relun 4651 |
The union of two relations is a relation. Compare Exercise 5 of
[TakeutiZaring] p. 25. (Contributed
by NM, 12-Aug-1994.)
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Theorem | relin1 4652 |
The intersection with a relation is a relation. (Contributed by NM,
16-Aug-1994.)
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Theorem | relin2 4653 |
The intersection with a relation is a relation. (Contributed by NM,
17-Jan-2006.)
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Theorem | reldif 4654 |
A difference cutting down a relation is a relation. (Contributed by NM,
31-Mar-1998.)
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Theorem | reliun 4655 |
An indexed union is a relation iff each member of its indexed family is
a relation. (Contributed by NM, 19-Dec-2008.)
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Theorem | reliin 4656 |
An indexed intersection is a relation if at least one of the member of the
indexed family is a relation. (Contributed by NM, 8-Mar-2014.)
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Theorem | reluni 4657* |
The union of a class is a relation iff any member is a relation.
Exercise 6 of [TakeutiZaring] p.
25 and its converse. (Contributed by
NM, 13-Aug-2004.)
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Theorem | relint 4658* |
The intersection of a class is a relation if at least one member is a
relation. (Contributed by NM, 8-Mar-2014.)
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Theorem | rel0 4659 |
The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
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Theorem | relopabi 4660 |
A class of ordered pairs is a relation. (Contributed by Mario Carneiro,
21-Dec-2013.)
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Theorem | relopab 4661 |
A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.)
(Unnecessary distinct variable restrictions were removed by Alan Sare,
9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)
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Theorem | mptrel 4662 |
The maps-to notation always describes a relationship. (Contributed by
Scott Fenton, 16-Apr-2012.)
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Theorem | reli 4663 |
The identity relation is a relation. Part of Exercise 4.12(p) of
[Mendelson] p. 235. (Contributed by
NM, 26-Apr-1998.) (Revised by
Mario Carneiro, 21-Dec-2013.)
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Theorem | rele 4664 |
The membership relation is a relation. (Contributed by NM,
26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
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Theorem | opabid2 4665* |
A relation expressed as an ordered pair abstraction. (Contributed by
NM, 11-Dec-2006.)
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Theorem | inopab 4666* |
Intersection of two ordered pair class abstractions. (Contributed by
NM, 30-Sep-2002.)
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Theorem | difopab 4667* |
The difference of two ordered-pair abstractions. (Contributed by Stefan
O'Rear, 17-Jan-2015.)
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Theorem | inxp 4668 |
The intersection of two cross products. Exercise 9 of [TakeutiZaring]
p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | xpindi 4669 |
Distributive law for cross product over intersection. Theorem 102 of
[Suppes] p. 52. (Contributed by NM,
26-Sep-2004.)
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Theorem | xpindir 4670 |
Distributive law for cross product over intersection. Similar to
Theorem 102 of [Suppes] p. 52.
(Contributed by NM, 26-Sep-2004.)
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Theorem | xpiindim 4671* |
Distributive law for cross product over indexed intersection.
(Contributed by Jim Kingdon, 7-Dec-2018.)
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Theorem | xpriindim 4672* |
Distributive law for cross product over relativized indexed
intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
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Theorem | eliunxp 4673* |
Membership in a union of cross products. Analogue of elxp 4551
for
nonconstant . (Contributed by Mario Carneiro,
29-Dec-2014.)
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Theorem | opeliunxp2 4674* |
Membership in a union of cross products. (Contributed by Mario
Carneiro, 14-Feb-2015.)
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Theorem | raliunxp 4675* |
Write a double restricted quantification as one universal quantifier.
In this version of ralxp 4677, is not assumed to be constant.
(Contributed by Mario Carneiro, 29-Dec-2014.)
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Theorem | rexiunxp 4676* |
Write a double restricted quantification as one universal quantifier.
In this version of rexxp 4678, is not assumed to be constant.
(Contributed by Mario Carneiro, 14-Feb-2015.)
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Theorem | ralxp 4677* |
Universal quantification restricted to a cross product is equivalent to
a double restricted quantification. The hypothesis specifies an
implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by
Mario Carneiro, 29-Dec-2014.)
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Theorem | rexxp 4678* |
Existential quantification restricted to a cross product is equivalent
to a double restricted quantification. (Contributed by NM,
11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)
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Theorem | djussxp 4679* |
Disjoint union is a subset of a cross product. (Contributed by Stefan
O'Rear, 21-Nov-2014.)
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Theorem | ralxpf 4680* |
Version of ralxp 4677 with bound-variable hypotheses. (Contributed
by NM,
18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | rexxpf 4681* |
Version of rexxp 4678 with bound-variable hypotheses. (Contributed
by NM,
19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | iunxpf 4682* |
Indexed union on a cross product is equals a double indexed union. The
hypothesis specifies an implicit substitution. (Contributed by NM,
19-Dec-2008.)
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Theorem | opabbi2dv 4683* |
Deduce equality of a relation and an ordered-pair class builder.
Compare abbi2dv 2256. (Contributed by NM, 24-Feb-2014.)
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Theorem | relop 4684* |
A necessary and sufficient condition for a Kuratowski ordered pair to be
a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this
detail.)
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Theorem | ideqg 4685 |
For sets, the identity relation is the same as equality. (Contributed
by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon,
27-Aug-2011.)
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Theorem | ideq 4686 |
For sets, the identity relation is the same as equality. (Contributed
by NM, 13-Aug-1995.)
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Theorem | ididg 4687 |
A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof
shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | issetid 4688 |
Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario
Carneiro, 26-Apr-2015.)
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Theorem | coss1 4689 |
Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
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Theorem | coss2 4690 |
Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
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Theorem | coeq1 4691 |
Equality theorem for composition of two classes. (Contributed by NM,
3-Jan-1997.)
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Theorem | coeq2 4692 |
Equality theorem for composition of two classes. (Contributed by NM,
3-Jan-1997.)
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Theorem | coeq1i 4693 |
Equality inference for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq2i 4694 |
Equality inference for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq1d 4695 |
Equality deduction for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq2d 4696 |
Equality deduction for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq12i 4697 |
Equality inference for composition of two classes. (Contributed by FL,
7-Jun-2012.)
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Theorem | coeq12d 4698 |
Equality deduction for composition of two classes. (Contributed by FL,
7-Jun-2012.)
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Theorem | nfco 4699 |
Bound-variable hypothesis builder for function value. (Contributed by
NM, 1-Sep-1999.)
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Theorem | elco 4700* |
Elements of a composed relation. (Contributed by BJ, 10-Jul-2022.)
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