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Type | Label | Description |
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Statement | ||
Theorem | elopabi 6101* | 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 6102* | 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 6103* | 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 6104* | 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 6105* | 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 6106* |
Functionality, domain and codomain of a class given by the maps-to
notation, where ![]() ![]() ![]() ![]() ![]() |
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Theorem | fmpo 6107* | 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 6108* | Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
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Theorem | mpofvex 6109* | 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 6110* | Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
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Theorem | dmmpo 6111* | Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
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Theorem | mpofvexi 6112* | 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 6113* | An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.) |
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Theorem | dmmpoga 6114* | Domain of an operation given by the maps-to notation, closed form of dmmpo 6111. (Contributed by Alexander van der Vekens, 10-Feb-2019.) |
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Theorem | dmmpog 6115* | Domain of an operation given by the maps-to notation, closed form of dmmpo 6111. 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 6116* | Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.) |
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Theorem | mpoexg 6117* | 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 6118* | 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 | mpoex 6119* | 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 6120* | 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 6121* | Composition of two functions. Variation of fmptco 5594 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.) |
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Theorem | oprabco 6122* | 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 6123* | Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.) |
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Theorem | df1st2 6124* |
An alternate possible definition of the ![]() |
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Theorem | df2nd2 6125* |
An alternate possible definition of the ![]() |
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Theorem | 1stconst 6126 |
The mapping of a restriction of the ![]() |
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Theorem | 2ndconst 6127 |
The mapping of a restriction of the ![]() |
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Theorem | dfmpo 6128* |
Alternate definition for the maps-to notation df-mpo 5787 (although it
requires that ![]() |
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Theorem | cnvf1olem 6129 | Lemma for cnvf1o 6130. (Contributed by Mario Carneiro, 27-Apr-2014.) |
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Theorem | cnvf1o 6130* | Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) |
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Theorem | f2ndf 6131 |
The ![]() ![]() ![]() ![]() |
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Theorem | fo2ndf 6132 |
The ![]() ![]() ![]() ![]() |
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Theorem | f1o2ndf1 6133 |
The ![]() ![]() ![]() ![]() |
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Theorem | algrflem 6134 | Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
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Theorem | algrflemg 6135 | Lemma for algrf 11762 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.) |
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Theorem | xporderlem 6136* | Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.) |
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Theorem | poxp 6137* | A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) |
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Theorem | spc2ed 6138* | Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.) |
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Theorem | cnvoprab 6139* | The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
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Theorem | f1od2 6140* | Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
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Theorem | disjxp1 6141* | The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | disjsnxp 6142* | The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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The following theorems are about maps-to operations (see df-mpo 5787) where the domain of the second argument depends on the domain of the first argument, especially when the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpox" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpox 5857, ovmpox 5907 and fmpox 6106). If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpoxop", and the maps-to operations are called "x-op maps-to operations" for short. | ||
Theorem | opeliunxp2f 6143* |
Membership in a union of Cartesian products, using bound-variable
hypothesis for ![]() |
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Theorem | mpoxopn0yelv 6144* | If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
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Theorem | mpoxopoveq 6145* | Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.) |
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Theorem | mpoxopovel 6146* | Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.) |
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Theorem | rbropapd 6147* | Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
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Theorem | rbropap 6148* |
Properties of a pair in a restricted binary relation ![]() ![]() ![]() ![]() |
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Syntax | ctpos 6149 | The transposition of a function. |
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Definition | df-tpos 6150* |
Define the transposition of a function, which is a function
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Theorem | tposss 6151 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposeq 6152 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposeqd 6153 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
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Theorem | tposssxp 6154 | The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
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Theorem | reltpos 6155 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | brtpos2 6156 |
Value of the transposition at a pair ![]() ![]() ![]() ![]() ![]() |
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Theorem | brtpos0 6157 | The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | reldmtpos 6158 |
Necessary and sufficient condition for ![]() ![]() |
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Theorem | brtposg 6159 | The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.) |
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Theorem | ottposg 6160 | The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.) |
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Theorem | dmtpos 6161 |
The domain of tpos ![]() ![]() ![]() |
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Theorem | rntpos 6162 |
The range of tpos ![]() ![]() ![]() |
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Theorem | tposexg 6163 | The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | ovtposg 6164 |
The transposition swaps the arguments in a two-argument function. When
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Theorem | tposfun 6165 | The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | dftpos2 6166* |
Alternate definition of tpos when ![]() |
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Theorem | dftpos3 6167* |
Alternate definition of tpos when ![]() |
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Theorem | dftpos4 6168* | Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tpostpos 6169 |
Value of the double transposition for a general class ![]() |
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Theorem | tpostpos2 6170 | Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
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Theorem | tposfn2 6171 | The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfo2 6172 | Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf2 6173 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf12 6174 | Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf1o2 6175 | Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfo 6176 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf 6177 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfn 6178 | Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tpos0 6179 | Transposition of the empty set. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposco 6180 | Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tpossym 6181* | Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tposeqi 6182 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposex 6183 | A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | nftpos 6184 | Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposoprab 6185* | Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposmpo 6186* | Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | pwuninel2 6187 | The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
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Theorem | 2pwuninelg 6188 | The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Jim Kingdon, 14-Jan-2020.) |
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Theorem | iunon 6189* |
The indexed union of a set of ordinal numbers ![]() ![]() ![]() ![]() |
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Syntax | wsmo 6190 | Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals. |
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Definition | df-smo 6191* | Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.) |
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Theorem | dfsmo2 6192* | Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.) |
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Theorem | issmo 6193* |
Conditions for which ![]() |
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Theorem | issmo2 6194* | Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.) |
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Theorem | smoeq 6195 | Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.) |
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Theorem | smodm 6196 | The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.) |
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Theorem | smores 6197 | A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
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Theorem | smores3 6198 | A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.) |
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Theorem | smores2 6199 | A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) |
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Theorem | smodm2 6200 | The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.) |
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