Theorem List for Intuitionistic Logic Explorer - 4701-4800 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | brcog 4701* |
Ordered pair membership in a composition. (Contributed by NM,
24-Feb-2015.)
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Theorem | opelco2g 4702* |
Ordered pair membership in a composition. (Contributed by NM,
27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
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Theorem | brcogw 4703 |
Ordered pair membership in a composition. (Contributed by Thierry
Arnoux, 14-Jan-2018.)
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Theorem | eqbrrdva 4704* |
Deduction from extensionality principle for relations, given an
equivalence only on the relation's domain and range. (Contributed by
Thierry Arnoux, 28-Nov-2017.)
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Theorem | brco 4705* |
Binary relation on a composition. (Contributed by NM, 21-Sep-2004.)
(Revised by Mario Carneiro, 24-Feb-2015.)
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Theorem | opelco 4706* |
Ordered pair membership in a composition. (Contributed by NM,
27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
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Theorem | cnvss 4707 |
Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
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Theorem | cnveq 4708 |
Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
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Theorem | cnveqi 4709 |
Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
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Theorem | cnveqd 4710 |
Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
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Theorem | elcnv 4711* |
Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed
by NM, 24-Mar-1998.)
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Theorem | elcnv2 4712* |
Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed
by NM, 11-Aug-2004.)
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Theorem | nfcnv 4713 |
Bound-variable hypothesis builder for converse. (Contributed by NM,
31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | opelcnvg 4714 |
Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | brcnvg 4715 |
The converse of a binary relation swaps arguments. Theorem 11 of [Suppes]
p. 61. (Contributed by NM, 10-Oct-2005.)
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Theorem | opelcnv 4716 |
Ordered-pair membership in converse. (Contributed by NM,
13-Aug-1995.)
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Theorem | brcnv 4717 |
The converse of a binary relation swaps arguments. Theorem 11 of
[Suppes] p. 61. (Contributed by NM,
13-Aug-1995.)
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Theorem | csbcnvg 4718 |
Move class substitution in and out of the converse of a function.
(Contributed by Thierry Arnoux, 8-Feb-2017.)
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Theorem | cnvco 4719 |
Distributive law of converse over class composition. Theorem 26 of
[Suppes] p. 64. (Contributed by NM,
19-Mar-1998.) (Proof shortened by
Andrew Salmon, 27-Aug-2011.)
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Theorem | cnvuni 4720* |
The converse of a class union is the (indexed) union of the converses of
its members. (Contributed by NM, 11-Aug-2004.)
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Theorem | dfdm3 4721* |
Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring]
p. 24. (Contributed by NM, 28-Dec-1996.)
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Theorem | dfrn2 4722* |
Alternate definition of range. Definition 4 of [Suppes] p. 60.
(Contributed by NM, 27-Dec-1996.)
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Theorem | dfrn3 4723* |
Alternate definition of range. Definition 6.5(2) of [TakeutiZaring]
p. 24. (Contributed by NM, 28-Dec-1996.)
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Theorem | elrn2g 4724* |
Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
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Theorem | elrng 4725* |
Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
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Theorem | dfdm4 4726 |
Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
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Theorem | dfdmf 4727* |
Definition of domain, using bound-variable hypotheses instead of
distinct variable conditions. (Contributed by NM, 8-Mar-1995.)
(Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | csbdmg 4728 |
Distribute proper substitution through the domain of a class.
(Contributed by Jim Kingdon, 8-Dec-2018.)
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Theorem | eldmg 4729* |
Domain membership. Theorem 4 of [Suppes] p. 59.
(Contributed by Mario
Carneiro, 9-Jul-2014.)
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Theorem | eldm2g 4730* |
Domain membership. Theorem 4 of [Suppes] p. 59.
(Contributed by NM,
27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
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Theorem | eldm 4731* |
Membership in a domain. Theorem 4 of [Suppes]
p. 59. (Contributed by
NM, 2-Apr-2004.)
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Theorem | eldm2 4732* |
Membership in a domain. Theorem 4 of [Suppes]
p. 59. (Contributed by
NM, 1-Aug-1994.)
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Theorem | dmss 4733 |
Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
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Theorem | dmeq 4734 |
Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
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Theorem | dmeqi 4735 |
Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
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Theorem | dmeqd 4736 |
Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
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Theorem | opeldm 4737 |
Membership of first of an ordered pair in a domain. (Contributed by NM,
30-Jul-1995.)
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Theorem | breldm 4738 |
Membership of first of a binary relation in a domain. (Contributed by
NM, 30-Jul-1995.)
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Theorem | opeldmg 4739 |
Membership of first of an ordered pair in a domain. (Contributed by Jim
Kingdon, 9-Jul-2019.)
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Theorem | breldmg 4740 |
Membership of first of a binary relation in a domain. (Contributed by
NM, 21-Mar-2007.)
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Theorem | dmun 4741 |
The domain of a union is the union of domains. Exercise 56(a) of
[Enderton] p. 65. (Contributed by NM,
12-Aug-1994.) (Proof shortened
by Andrew Salmon, 27-Aug-2011.)
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Theorem | dmin 4742 |
The domain of an intersection belong to the intersection of domains.
Theorem 6 of [Suppes] p. 60.
(Contributed by NM, 15-Sep-2004.)
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Theorem | dmiun 4743 |
The domain of an indexed union. (Contributed by Mario Carneiro,
26-Apr-2016.)
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Theorem | dmuni 4744* |
The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
(Contributed by NM, 3-Feb-2004.)
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Theorem | dmopab 4745* |
The domain of a class of ordered pairs. (Contributed by NM,
16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
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Theorem | dmopabss 4746* |
Upper bound for the domain of a restricted class of ordered pairs.
(Contributed by NM, 31-Jan-2004.)
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Theorem | dmopab3 4747* |
The domain of a restricted class of ordered pairs. (Contributed by NM,
31-Jan-2004.)
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Theorem | dm0 4748 |
The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1]
p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | dmi 4749 |
The domain of the identity relation is the universe. (Contributed by
NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | dmv 4750 |
The domain of the universe is the universe. (Contributed by NM,
8-Aug-2003.)
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Theorem | dm0rn0 4751 |
An empty domain implies an empty range. For a similar theorem for
whether the domain and range are inhabited, see dmmrnm 4753. (Contributed
by NM, 21-May-1998.)
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Theorem | reldm0 4752 |
A relation is empty iff its domain is empty. (Contributed by NM,
15-Sep-2004.)
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Theorem | dmmrnm 4753* |
A domain is inhabited if and only if the range is inhabited.
(Contributed by Jim Kingdon, 15-Dec-2018.)
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Theorem | dmxpm 4754* |
The domain of a cross product. Part of Theorem 3.13(x) of [Monk1]
p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | dmxpid 4755 |
The domain of a square Cartesian product. (Contributed by NM,
28-Jul-1995.) (Revised by Jim Kingdon, 11-Apr-2023.)
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Theorem | dmxpin 4756 |
The domain of the intersection of two square Cartesian products. Unlike
dmin 4742, equality holds. (Contributed by NM,
29-Jan-2008.)
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Theorem | xpid11 4757 |
The Cartesian product of a class with itself is one-to-one. (Contributed
by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | dmcnvcnv 4758 |
The domain of the double converse of a class (which doesn't have to be a
relation as in dfrel2 4984). (Contributed by NM, 8-Apr-2007.)
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Theorem | rncnvcnv 4759 |
The range of the double converse of a class. (Contributed by NM,
8-Apr-2007.)
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Theorem | elreldm 4760 |
The first member of an ordered pair in a relation belongs to the domain
of the relation. (Contributed by NM, 28-Jul-2004.)
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Theorem | rneq 4761 |
Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
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Theorem | rneqi 4762 |
Equality inference for range. (Contributed by NM, 4-Mar-2004.)
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Theorem | rneqd 4763 |
Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
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Theorem | rnss 4764 |
Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
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Theorem | brelrng 4765 |
The second argument of a binary relation belongs to its range.
(Contributed by NM, 29-Jun-2008.)
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Theorem | opelrng 4766 |
Membership of second member of an ordered pair in a range. (Contributed
by Jim Kingdon, 26-Jan-2019.)
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Theorem | brelrn 4767 |
The second argument of a binary relation belongs to its range.
(Contributed by NM, 13-Aug-2004.)
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Theorem | opelrn 4768 |
Membership of second member of an ordered pair in a range. (Contributed
by NM, 23-Feb-1997.)
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Theorem | releldm 4769 |
The first argument of a binary relation belongs to its domain.
(Contributed by NM, 2-Jul-2008.)
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Theorem | relelrn 4770 |
The second argument of a binary relation belongs to its range.
(Contributed by NM, 2-Jul-2008.)
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Theorem | releldmb 4771* |
Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
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Theorem | relelrnb 4772* |
Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
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Theorem | releldmi 4773 |
The first argument of a binary relation belongs to its domain.
(Contributed by NM, 28-Apr-2015.)
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Theorem | relelrni 4774 |
The second argument of a binary relation belongs to its range.
(Contributed by NM, 28-Apr-2015.)
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Theorem | dfrnf 4775* |
Definition of range, using bound-variable hypotheses instead of distinct
variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by
Mario Carneiro, 15-Oct-2016.)
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Theorem | elrn2 4776* |
Membership in a range. (Contributed by NM, 10-Jul-1994.)
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Theorem | elrn 4777* |
Membership in a range. (Contributed by NM, 2-Apr-2004.)
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Theorem | nfdm 4778 |
Bound-variable hypothesis builder for domain. (Contributed by NM,
30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | nfrn 4779 |
Bound-variable hypothesis builder for range. (Contributed by NM,
1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | dmiin 4780 |
Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
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Theorem | rnopab 4781* |
The range of a class of ordered pairs. (Contributed by NM,
14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
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Theorem | rnmpt 4782* |
The range of a function in maps-to notation. (Contributed by Scott
Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
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Theorem | elrnmpt 4783* |
The range of a function in maps-to notation. (Contributed by Mario
Carneiro, 20-Feb-2015.)
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Theorem | elrnmpt1s 4784* |
Elementhood in an image set. (Contributed by Mario Carneiro,
12-Sep-2015.)
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Theorem | elrnmpt1 4785 |
Elementhood in an image set. (Contributed by Mario Carneiro,
31-Aug-2015.)
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Theorem | elrnmptg 4786* |
Membership in the range of a function. (Contributed by NM,
27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
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Theorem | elrnmpti 4787* |
Membership in the range of a function. (Contributed by NM,
30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
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Theorem | elrnmptdv 4788* |
Elementhood in the range of a function in maps-to notation, deduction
form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
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Theorem | elrnmpt2d 4789* |
Elementhood in the range of a function in maps-to notation, deduction
form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
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Theorem | rn0 4790 |
The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1]
p. 36. (Contributed by NM, 4-Jul-1994.)
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Theorem | dfiun3g 4791 |
Alternate definition of indexed union when is a set. (Contributed
by Mario Carneiro, 31-Aug-2015.)
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Theorem | dfiin3g 4792 |
Alternate definition of indexed intersection when is a set.
(Contributed by Mario Carneiro, 31-Aug-2015.)
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Theorem | dfiun3 4793 |
Alternate definition of indexed union when is a set. (Contributed
by Mario Carneiro, 31-Aug-2015.)
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Theorem | dfiin3 4794 |
Alternate definition of indexed intersection when is a set.
(Contributed by Mario Carneiro, 31-Aug-2015.)
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Theorem | riinint 4795* |
Express a relative indexed intersection as an intersection.
(Contributed by Stefan O'Rear, 22-Feb-2015.)
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Theorem | relrn0 4796 |
A relation is empty iff its range is empty. (Contributed by NM,
15-Sep-2004.)
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Theorem | dmrnssfld 4797 |
The domain and range of a class are included in its double union.
(Contributed by NM, 13-May-2008.)
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Theorem | dmexg 4798 |
The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26.
(Contributed by NM, 7-Apr-1995.)
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Theorem | rnexg 4799 |
The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26.
Similar to Lemma 3D of [Enderton] p. 41.
(Contributed by NM,
31-Mar-1995.)
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Theorem | dmex 4800 |
The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring]
p. 26. (Contributed by NM, 7-Jul-2008.)
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