Theorem List for Intuitionistic Logic Explorer - 6101-6200 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | opabex3 6101* |
Existence of an ordered pair abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iunex 6102* |
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 6103* |
Existence of an existentially restricted class abstraction. is
normally has free-variable parameters and . See also
abrexex 6096. (Contributed by NM, 12-Sep-2004.)
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Theorem | abexssex 6104* |
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 6105* |
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 6106* |
Existence of an operator abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | oprabex 6107* |
Existence of an operation class abstraction. (Contributed by NM,
19-Oct-2004.)
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Theorem | oprabex3 6108* |
Existence of an operation class abstraction (special case).
(Contributed by NM, 19-Oct-2004.)
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Theorem | oprabrexex2 6109* |
Existence of an existentially restricted operation abstraction.
(Contributed by Jeff Madsen, 11-Jun-2010.)
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Theorem | ab2rexex 6110* |
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 6096. (Contributed by NM,
20-Sep-2011.)
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Theorem | ab2rexex2 6111* |
Existence of an existentially restricted class abstraction.
normally has free-variable parameters , , and .
Compare abrexex2 6103. (Contributed by NM, 20-Sep-2011.)
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Theorem | xpexgALT 6112 |
The cross product of two sets is a set. Proposition 6.2 of
[TakeutiZaring] p. 23. This
version is proven using Replacement; see
xpexg 4725 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 6113* |
General value of with no assumptions on functionality
of and . (Contributed by Stefan
O'Rear, 24-Jan-2015.)
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Theorem | offres 6114 |
Pointwise combination commutes with restriction. (Contributed by Stefan
O'Rear, 24-Jan-2015.)
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Theorem | ofmres 6115* |
Equivalent expressions for a restriction of the function operation map.
Unlike which is a proper class,
can
be a set by ofmresex 6116, allowing it to be used as a function or
structure argument. By ofmresval 6072, 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 6116 |
Existence of a restriction of the function operation map. (Contributed
by NM, 20-Oct-2014.)
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2.6.15 First and second members of an ordered
pair
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Syntax | c1st 6117 |
Extend the definition of a class to include the first member an ordered
pair function.
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Syntax | c2nd 6118 |
Extend the definition of a class to include the second member an ordered
pair function.
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Definition | df-1st 6119 |
Define a function that extracts the first member, or abscissa, of an
ordered pair. Theorem op1st 6125 proves that it does this. For example,
( 3 , 4 ) = 3 . Equivalent to Definition
5.13 (i) of
[Monk1] p. 52 (compare op1sta 5092 and op1stb 4463). The notation is the same
as Monk's. (Contributed by NM, 9-Oct-2004.)
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Definition | df-2nd 6120 |
Define a function that extracts the second member, or ordinate, of an
ordered pair. Theorem op2nd 6126 proves that it does this. For example,
3 , 4 ) = 4 . Equivalent to Definition 5.13 (ii)
of [Monk1] p. 52 (compare op2nda 5095 and op2ndb 5094). The notation is the
same as Monk's. (Contributed by NM, 9-Oct-2004.)
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Theorem | 1stvalg 6121 |
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 6122 |
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 6123 |
The value of the first-member function at the empty set. (Contributed by
NM, 23-Apr-2007.)
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Theorem | 2nd0 6124 |
The value of the second-member function at the empty set. (Contributed by
NM, 23-Apr-2007.)
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Theorem | op1st 6125 |
Extract the first member of an ordered pair. (Contributed by NM,
5-Oct-2004.)
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Theorem | op2nd 6126 |
Extract the second member of an ordered pair. (Contributed by NM,
5-Oct-2004.)
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Theorem | op1std 6127 |
Extract the first member of an ordered pair. (Contributed by Mario
Carneiro, 31-Aug-2015.)
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Theorem | op2ndd 6128 |
Extract the second member of an ordered pair. (Contributed by Mario
Carneiro, 31-Aug-2015.)
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Theorem | op1stg 6129 |
Extract the first member of an ordered pair. (Contributed by NM,
19-Jul-2005.)
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Theorem | op2ndg 6130 |
Extract the second member of an ordered pair. (Contributed by NM,
19-Jul-2005.)
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Theorem | ot1stg 6131 |
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 6131,
ot2ndg 6132, ot3rdgg 6133.) (Contributed by NM, 3-Apr-2015.) (Revised
by
Mario Carneiro, 2-May-2015.)
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Theorem | ot2ndg 6132 |
Extract the second member of an ordered triple. (See ot1stg 6131 comment.)
(Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro,
2-May-2015.)
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Theorem | ot3rdgg 6133 |
Extract the third member of an ordered triple. (See ot1stg 6131 comment.)
(Contributed by NM, 3-Apr-2015.)
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Theorem | 1stval2 6134 |
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 6135 |
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 6136 |
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 6137 |
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 6138 |
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 6139 |
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 6140* |
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 6141* |
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 6142 |
Composition of the first member function with another function.
(Contributed by NM, 12-Oct-2007.)
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Theorem | 2ndcof 6143 |
Composition of the second member function with another function.
(Contributed by FL, 15-Oct-2012.)
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Theorem | xp1st 6144 |
Location of the first element of a Cartesian product. (Contributed by
Jeff Madsen, 2-Sep-2009.)
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Theorem | xp2nd 6145 |
Location of the second element of a Cartesian product. (Contributed by
Jeff Madsen, 2-Sep-2009.)
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Theorem | 1stexg 6146 |
Existence of the first member of a set. (Contributed by Jim Kingdon,
26-Jan-2019.)
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Theorem | 2ndexg 6147 |
Existence of the first member of a set. (Contributed by Jim Kingdon,
26-Jan-2019.)
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Theorem | elxp6 6148 |
Membership in a cross product. This version requires no quantifiers or
dummy variables. See also elxp4 5098. (Contributed by NM, 9-Oct-2004.)
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Theorem | elxp7 6149 |
Membership in a cross product. This version requires no quantifiers or
dummy variables. See also elxp4 5098. (Contributed by NM, 19-Aug-2006.)
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Theorem | oprssdmm 6150* |
Domain of closure of an operation. (Contributed by Jim Kingdon,
23-Oct-2023.)
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Theorem | eqopi 6151 |
Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.)
(Revised by Mario Carneiro, 23-Feb-2014.)
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Theorem | xp2 6152* |
Representation of cross product based on ordered pair component
functions. (Contributed by NM, 16-Sep-2006.)
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Theorem | unielxp 6153 |
The membership relation for a cross product is inherited by union.
(Contributed by NM, 16-Sep-2006.)
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Theorem | 1st2nd2 6154 |
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 6155 |
An ordered pair theorem for members of cross products. (Contributed by
NM, 20-Jun-2007.)
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Theorem | eqop 6156 |
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 6157 |
Two ways to express equality with an ordered pair. (Contributed by NM,
25-Feb-2014.)
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Theorem | op1steq 6158* |
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 6159 |
Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
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Theorem | 1st2nd 6160 |
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 6161 |
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 6162 |
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 6163 |
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 6164* |
Two ways of expressing membership in the domain of a relation.
(Contributed by NM, 22-Sep-2013.)
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Theorem | reldm 6165* |
An expression for the domain of a relation. (Contributed by NM,
22-Sep-2013.)
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Theorem | sbcopeq1a 6166 |
Equality theorem for substitution of a class for an ordered pair (analog
of sbceq1a 2964 that avoids the existential quantifiers of copsexg 4229).
(Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro,
31-Aug-2015.)
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Theorem | csbopeq1a 6167 |
Equality theorem for substitution of a class for an ordered pair
in (analog of csbeq1a 3058). (Contributed by NM,
19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
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Theorem | dfopab2 6168* |
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 6169* |
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 6170* |
Operation class abstraction expressed without existential quantifiers.
(Contributed by NM, 16-Dec-2008.)
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Theorem | dfoprab4 6171* |
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 6172* |
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 6173* |
Define the cross product of three classes. Compare df-xp 4617.
(Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro,
3-Nov-2015.)
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Theorem | elopabi 6174* |
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 6175* |
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 6176* |
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 6177* |
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 6178* |
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 6179* |
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 6180* |
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 6181* |
Functionality and domain of a class given by the maps-to notation.
(Contributed by FL, 17-May-2010.)
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Theorem | mpofvex 6182* |
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 6183* |
Functionality and domain of a class given by the maps-to notation.
(Contributed by FL, 17-May-2010.)
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Theorem | dmmpo 6184* |
Domain of a class given by the maps-to notation. (Contributed by FL,
17-May-2010.)
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Theorem | mpofvexi 6185* |
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 6186* |
An operation's value belongs to its range. (Contributed by AV,
27-Jan-2020.)
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Theorem | dmmpoga 6187* |
Domain of an operation given by the maps-to notation, closed form of
dmmpo 6184. (Contributed by Alexander van der Vekens,
10-Feb-2019.)
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Theorem | dmmpog 6188* |
Domain of an operation given by the maps-to notation, closed form of
dmmpo 6184. 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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Theorem | mpoexxg 6189* |
Existence of an operation class abstraction (version for dependent
domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
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Theorem | mpoexg 6190* |
Existence of an operation class abstraction (special case).
(Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro,
1-Sep-2015.)
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Theorem | mpoexga 6191* |
If the domain of an operation given by maps-to notation is a set, the
operation is a set. (Contributed by NM, 12-Sep-2011.)
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Theorem | mpoexw 6192* |
Weak version of mpoex 6193 that holds without ax-coll 4104. If the domain
and codomain of an operation given by maps-to notation are sets, the
operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.)
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Theorem | mpoex 6193* |
If the domain of an operation given by maps-to notation is a set, the
operation is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)
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Theorem | fnmpoovd 6194* |
A function with a Cartesian product as domain is a mapping with two
arguments defined by its operation values. (Contributed by AV,
20-Feb-2019.) (Revised by AV, 3-Jul-2022.)
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Theorem | fmpoco 6195* |
Composition of two functions. Variation of fmptco 5662 when the second
function has two arguments. (Contributed by Mario Carneiro,
8-Feb-2015.)
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Theorem | oprabco 6196* |
Composition of a function with an operator abstraction. (Contributed by
Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro,
26-Sep-2015.)
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Theorem | oprab2co 6197* |
Composition of operator abstractions. (Contributed by Jeff Madsen,
2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
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Theorem | df1st2 6198* |
An alternate possible definition of the function. (Contributed
by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
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Theorem | df2nd2 6199* |
An alternate possible definition of the function. (Contributed
by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
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Theorem | 1stconst 6200 |
The mapping of a restriction of the function to a constant
function. (Contributed by NM, 14-Dec-2008.)
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