Theorem List for Intuitionistic Logic Explorer - 6001-6100 *Has distinct variable
group(s)
Type | Label | Description |
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2.6.13 Functions (continued)
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Theorem | resfunexgALT 6001 |
The restriction of a function to a set exists. Compare Proposition 6.17
of [TakeutiZaring] p. 28. This
version has a shorter proof than
resfunexg 5634 but requires ax-pow 4093 and ax-un 4350. (Contributed by NM,
7-Apr-1995.) (Proof modification is discouraged.)
(New usage is discouraged.)
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Theorem | cofunexg 6002 |
Existence of a composition when the first member is a function.
(Contributed by NM, 8-Oct-2007.)
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Theorem | cofunex2g 6003 |
Existence of a composition when the second member is one-to-one.
(Contributed by NM, 8-Oct-2007.)
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Theorem | fnexALT 6004 |
If the domain of a function is a set, the function is a set. Theorem
6.16(1) of [TakeutiZaring] p. 28.
This theorem is derived using the Axiom
of Replacement in the form of funimaexg 5202. This version of fnex 5635
uses
ax-pow 4093 and ax-un 4350, whereas fnex 5635
does not. (Contributed by NM,
14-Aug-1994.) (Proof modification is discouraged.)
(New usage is discouraged.)
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Theorem | funrnex 6005 |
If the domain of a function exists, so does its range. Part of Theorem
4.15(v) of [Monk1] p. 46. This theorem is
derived using the Axiom of
Replacement in the form of funex 5636. (Contributed by NM, 11-Nov-1995.)
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Theorem | fornex 6006 |
If the domain of an onto function exists, so does its codomain.
(Contributed by NM, 23-Jul-2004.)
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Theorem | f1dmex 6007 |
If the codomain of a one-to-one function exists, so does its domain. This
can be thought of as a form of the Axiom of Replacement. (Contributed by
NM, 4-Sep-2004.)
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Theorem | abrexex 6008* |
Existence of a class abstraction of existentially restricted sets.
is normally a free-variable parameter in the class expression
substituted for , which can be thought of as . This
simple-looking theorem is actually quite powerful and appears to involve
the Axiom of Replacement in an intrinsic way, as can be seen by tracing
back through the path mptexg 5638, funex 5636, fnex 5635, resfunexg 5634, and
funimaexg 5202. See also abrexex2 6015. (Contributed by NM, 16-Oct-2003.)
(Proof shortened by Mario Carneiro, 31-Aug-2015.)
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Theorem | abrexexg 6009* |
Existence of a class abstraction of existentially restricted sets.
is normally a free-variable parameter in . The antecedent assures
us that is a
set. (Contributed by NM, 3-Nov-2003.)
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Theorem | iunexg 6010* |
The existence of an indexed union. is normally a free-variable
parameter in .
(Contributed by NM, 23-Mar-2006.)
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Theorem | abrexex2g 6011* |
Existence of an existentially restricted class abstraction.
(Contributed by Jeff Madsen, 2-Sep-2009.)
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Theorem | opabex3d 6012* |
Existence of an ordered pair abstraction, deduction version.
(Contributed by Alexander van der Vekens, 19-Oct-2017.)
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Theorem | opabex3 6013* |
Existence of an ordered pair abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iunex 6014* |
The existence of an indexed union. is normally a free-variable
parameter in the class expression substituted for , which can be
read informally as . (Contributed by NM, 13-Oct-2003.)
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Theorem | abrexex2 6015* |
Existence of an existentially restricted class abstraction. is
normally has free-variable parameters and . See also
abrexex 6008. (Contributed by NM, 12-Sep-2004.)
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Theorem | abexssex 6016* |
Existence of a class abstraction with an existentially quantified
expression. Both and can be
free in .
(Contributed
by NM, 29-Jul-2006.)
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Theorem | abexex 6017* |
A condition where a class builder continues to exist after its wff is
existentially quantified. (Contributed by NM, 4-Mar-2007.)
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Theorem | oprabexd 6018* |
Existence of an operator abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | oprabex 6019* |
Existence of an operation class abstraction. (Contributed by NM,
19-Oct-2004.)
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Theorem | oprabex3 6020* |
Existence of an operation class abstraction (special case).
(Contributed by NM, 19-Oct-2004.)
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Theorem | oprabrexex2 6021* |
Existence of an existentially restricted operation abstraction.
(Contributed by Jeff Madsen, 11-Jun-2010.)
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Theorem | ab2rexex 6022* |
Existence of a class abstraction of existentially restricted sets.
Variables and
are normally
free-variable parameters in the
class expression substituted for , which can be thought of as
. See comments for abrexex 6008. (Contributed by NM,
20-Sep-2011.)
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Theorem | ab2rexex2 6023* |
Existence of an existentially restricted class abstraction.
normally has free-variable parameters , , and .
Compare abrexex2 6015. (Contributed by NM, 20-Sep-2011.)
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Theorem | xpexgALT 6024 |
The cross product of two sets is a set. Proposition 6.2 of
[TakeutiZaring] p. 23. This
version is proven using Replacement; see
xpexg 4648 for a version that uses the Power Set axiom
instead.
(Contributed by Mario Carneiro, 20-May-2013.)
(Proof modification is discouraged.) (New usage is discouraged.)
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Theorem | offval3 6025* |
General value of with no assumptions on functionality
of and . (Contributed by Stefan
O'Rear, 24-Jan-2015.)
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Theorem | offres 6026 |
Pointwise combination commutes with restriction. (Contributed by Stefan
O'Rear, 24-Jan-2015.)
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Theorem | ofmres 6027* |
Equivalent expressions for a restriction of the function operation map.
Unlike which is a proper class,
can
be a set by ofmresex 6028, allowing it to be used as a function or
structure argument. By ofmresval 5986, the restricted operation map
values are the same as the original values, allowing theorems for
to be reused. (Contributed by NM, 20-Oct-2014.)
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Theorem | ofmresex 6028 |
Existence of a restriction of the function operation map. (Contributed
by NM, 20-Oct-2014.)
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2.6.14 First and second members of an ordered
pair
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Syntax | c1st 6029 |
Extend the definition of a class to include the first member an ordered
pair function.
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Syntax | c2nd 6030 |
Extend the definition of a class to include the second member an ordered
pair function.
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Definition | df-1st 6031 |
Define a function that extracts the first member, or abscissa, of an
ordered pair. Theorem op1st 6037 proves that it does this. For example,
( 3 , 4 ) = 3 . Equivalent to Definition
5.13 (i) of
[Monk1] p. 52 (compare op1sta 5015 and op1stb 4394). The notation is the same
as Monk's. (Contributed by NM, 9-Oct-2004.)
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Definition | df-2nd 6032 |
Define a function that extracts the second member, or ordinate, of an
ordered pair. Theorem op2nd 6038 proves that it does this. For example,
3 , 4 ) = 4 . Equivalent to Definition 5.13 (ii)
of [Monk1] p. 52 (compare op2nda 5018 and op2ndb 5017). The notation is the
same as Monk's. (Contributed by NM, 9-Oct-2004.)
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Theorem | 1stvalg 6033 |
The value of the function that extracts the first member of an ordered
pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro,
8-Sep-2013.)
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Theorem | 2ndvalg 6034 |
The value of the function that extracts the second member of an ordered
pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro,
8-Sep-2013.)
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Theorem | 1st0 6035 |
The value of the first-member function at the empty set. (Contributed by
NM, 23-Apr-2007.)
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Theorem | 2nd0 6036 |
The value of the second-member function at the empty set. (Contributed by
NM, 23-Apr-2007.)
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Theorem | op1st 6037 |
Extract the first member of an ordered pair. (Contributed by NM,
5-Oct-2004.)
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Theorem | op2nd 6038 |
Extract the second member of an ordered pair. (Contributed by NM,
5-Oct-2004.)
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Theorem | op1std 6039 |
Extract the first member of an ordered pair. (Contributed by Mario
Carneiro, 31-Aug-2015.)
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Theorem | op2ndd 6040 |
Extract the second member of an ordered pair. (Contributed by Mario
Carneiro, 31-Aug-2015.)
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Theorem | op1stg 6041 |
Extract the first member of an ordered pair. (Contributed by NM,
19-Jul-2005.)
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Theorem | op2ndg 6042 |
Extract the second member of an ordered pair. (Contributed by NM,
19-Jul-2005.)
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Theorem | ot1stg 6043 |
Extract the first member of an ordered triple. (Due to infrequent
usage, it isn't worthwhile at this point to define special extractors
for triples, so we reuse the ordered pair extractors for ot1stg 6043,
ot2ndg 6044, ot3rdgg 6045.) (Contributed by NM, 3-Apr-2015.) (Revised
by
Mario Carneiro, 2-May-2015.)
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Theorem | ot2ndg 6044 |
Extract the second member of an ordered triple. (See ot1stg 6043 comment.)
(Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro,
2-May-2015.)
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Theorem | ot3rdgg 6045 |
Extract the third member of an ordered triple. (See ot1stg 6043 comment.)
(Contributed by NM, 3-Apr-2015.)
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Theorem | 1stval2 6046 |
Alternate value of the function that extracts the first member of an
ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by
NM, 18-Aug-2006.)
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Theorem | 2ndval2 6047 |
Alternate value of the function that extracts the second member of an
ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by
NM, 18-Aug-2006.)
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Theorem | fo1st 6048 |
The function
maps the universe onto the universe. (Contributed
by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
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Theorem | fo2nd 6049 |
The function
maps the universe onto the universe. (Contributed
by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
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Theorem | f1stres 6050 |
Mapping of a restriction of the (first member of an ordered
pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario
Carneiro, 8-Sep-2013.)
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Theorem | f2ndres 6051 |
Mapping of a restriction of the (second member of an ordered
pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario
Carneiro, 8-Sep-2013.)
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Theorem | fo1stresm 6052* |
Onto mapping of a restriction of the (first member of an ordered
pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
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Theorem | fo2ndresm 6053* |
Onto mapping of a restriction of the (second member of an
ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
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Theorem | 1stcof 6054 |
Composition of the first member function with another function.
(Contributed by NM, 12-Oct-2007.)
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Theorem | 2ndcof 6055 |
Composition of the second member function with another function.
(Contributed by FL, 15-Oct-2012.)
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Theorem | xp1st 6056 |
Location of the first element of a Cartesian product. (Contributed by
Jeff Madsen, 2-Sep-2009.)
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Theorem | xp2nd 6057 |
Location of the second element of a Cartesian product. (Contributed by
Jeff Madsen, 2-Sep-2009.)
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Theorem | 1stexg 6058 |
Existence of the first member of a set. (Contributed by Jim Kingdon,
26-Jan-2019.)
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Theorem | 2ndexg 6059 |
Existence of the first member of a set. (Contributed by Jim Kingdon,
26-Jan-2019.)
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Theorem | elxp6 6060 |
Membership in a cross product. This version requires no quantifiers or
dummy variables. See also elxp4 5021. (Contributed by NM, 9-Oct-2004.)
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Theorem | elxp7 6061 |
Membership in a cross product. This version requires no quantifiers or
dummy variables. See also elxp4 5021. (Contributed by NM, 19-Aug-2006.)
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Theorem | oprssdmm 6062* |
Domain of closure of an operation. (Contributed by Jim Kingdon,
23-Oct-2023.)
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Theorem | eqopi 6063 |
Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.)
(Revised by Mario Carneiro, 23-Feb-2014.)
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Theorem | xp2 6064* |
Representation of cross product based on ordered pair component
functions. (Contributed by NM, 16-Sep-2006.)
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Theorem | unielxp 6065 |
The membership relation for a cross product is inherited by union.
(Contributed by NM, 16-Sep-2006.)
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Theorem | 1st2nd2 6066 |
Reconstruction of a member of a cross product in terms of its ordered pair
components. (Contributed by NM, 20-Oct-2013.)
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Theorem | xpopth 6067 |
An ordered pair theorem for members of cross products. (Contributed by
NM, 20-Jun-2007.)
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Theorem | eqop 6068 |
Two ways to express equality with an ordered pair. (Contributed by NM,
3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
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Theorem | eqop2 6069 |
Two ways to express equality with an ordered pair. (Contributed by NM,
25-Feb-2014.)
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Theorem | op1steq 6070* |
Two ways of expressing that an element is the first member of an ordered
pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro,
23-Feb-2014.)
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Theorem | 2nd1st 6071 |
Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
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Theorem | 1st2nd 6072 |
Reconstruction of a member of a relation in terms of its ordered pair
components. (Contributed by NM, 29-Aug-2006.)
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Theorem | 1stdm 6073 |
The first ordered pair component of a member of a relation belongs to the
domain of the relation. (Contributed by NM, 17-Sep-2006.)
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Theorem | 2ndrn 6074 |
The second ordered pair component of a member of a relation belongs to the
range of the relation. (Contributed by NM, 17-Sep-2006.)
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Theorem | 1st2ndbr 6075 |
Express an element of a relation as a relationship between first and
second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
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Theorem | releldm2 6076* |
Two ways of expressing membership in the domain of a relation.
(Contributed by NM, 22-Sep-2013.)
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Theorem | reldm 6077* |
An expression for the domain of a relation. (Contributed by NM,
22-Sep-2013.)
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Theorem | sbcopeq1a 6078 |
Equality theorem for substitution of a class for an ordered pair (analog
of sbceq1a 2913 that avoids the existential quantifiers of copsexg 4161).
(Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro,
31-Aug-2015.)
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Theorem | csbopeq1a 6079 |
Equality theorem for substitution of a class for an ordered pair
in (analog of csbeq1a 3007). (Contributed by NM,
19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
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Theorem | dfopab2 6080* |
A way to define an ordered-pair class abstraction without using
existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by
Mario Carneiro, 31-Aug-2015.)
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Theorem | dfoprab3s 6081* |
A way to define an operation class abstraction without using existential
quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario
Carneiro, 31-Aug-2015.)
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Theorem | dfoprab3 6082* |
Operation class abstraction expressed without existential quantifiers.
(Contributed by NM, 16-Dec-2008.)
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Theorem | dfoprab4 6083* |
Operation class abstraction expressed without existential quantifiers.
(Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro,
31-Aug-2015.)
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Theorem | dfoprab4f 6084* |
Operation class abstraction expressed without existential quantifiers.
(Unnecessary distinct variable restrictions were removed by David
Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by
Mario Carneiro, 31-Aug-2015.)
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Theorem | dfxp3 6085* |
Define the cross product of three classes. Compare df-xp 4540.
(Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro,
3-Nov-2015.)
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Theorem | elopabi 6086* |
A consequence of membership in an ordered-pair class abstraction, using
ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
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Theorem | eloprabi 6087* |
A consequence of membership in an operation class abstraction, using
ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by
David Abernethy, 19-Jun-2012.)
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Theorem | mpomptsx 6088* |
Express a two-argument function as a one-argument function, or
vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
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Theorem | mpompts 6089* |
Express a two-argument function as a one-argument function, or
vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
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Theorem | dmmpossx 6090* |
The domain of a mapping is a subset of its base class. (Contributed by
Mario Carneiro, 9-Feb-2015.)
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Theorem | fmpox 6091* |
Functionality, domain and codomain of a class given by the maps-to
notation, where is not constant but depends on .
(Contributed by NM, 29-Dec-2014.)
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Theorem | fmpo 6092* |
Functionality, domain and range of a class given by the maps-to
notation. (Contributed by FL, 17-May-2010.)
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Theorem | fnmpo 6093* |
Functionality and domain of a class given by the maps-to notation.
(Contributed by FL, 17-May-2010.)
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Theorem | mpofvex 6094* |
Sufficient condition for an operation maps-to notation to be set-like.
(Contributed by Mario Carneiro, 3-Jul-2019.)
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Theorem | fnmpoi 6095* |
Functionality and domain of a class given by the maps-to notation.
(Contributed by FL, 17-May-2010.)
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Theorem | dmmpo 6096* |
Domain of a class given by the maps-to notation. (Contributed by FL,
17-May-2010.)
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Theorem | mpofvexi 6097* |
Sufficient condition for an operation maps-to notation to be set-like.
(Contributed by Mario Carneiro, 3-Jul-2019.)
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Theorem | ovmpoelrn 6098* |
An operation's value belongs to its range. (Contributed by AV,
27-Jan-2020.)
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Theorem | dmmpoga 6099* |
Domain of an operation given by the maps-to notation, closed form of
dmmpo 6096. (Contributed by Alexander van der Vekens,
10-Feb-2019.)
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Theorem | dmmpog 6100* |
Domain of an operation given by the maps-to notation, closed form of
dmmpo 6096. Caution: This theorem is only valid in the
very special case
where the value of the mapping is a constant! (Contributed by Alexander
van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.)
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