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Type | Label | Description |
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Statement | ||
Theorem | 1st2nd 6201 | 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 6202 | 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 6203 | 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 6204 | 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 6205* | Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
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Theorem | reldm 6206* | An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
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Theorem | sbcopeq1a 6207 | Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2987 that avoids the existential quantifiers of copsexg 4259). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
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Theorem | csbopeq1a 6208 |
Equality theorem for substitution of a class ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dfopab2 6209* | 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 6210* | 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 6211* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.) |
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Theorem | dfoprab4 6212* | 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 6213* | 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 6214* | Define the cross product of three classes. Compare df-xp 4647. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) |
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Theorem | elopabi 6215* | 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 6216* | 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 6217* | 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 6218* | 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 6219* | 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 6220* |
Functionality, domain and codomain of a class given by the maps-to
notation, where ![]() ![]() ![]() ![]() ![]() |
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Theorem | fmpo 6221* | 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 6222* | Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
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Theorem | mpofvex 6223* | 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 6224* | Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
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Theorem | dmmpo 6225* | Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
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Theorem | mpofvexi 6226* | 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 6227* | An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.) |
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Theorem | dmmpoga 6228* | Domain of an operation given by the maps-to notation, closed form of dmmpo 6225. (Contributed by Alexander van der Vekens, 10-Feb-2019.) |
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Theorem | dmmpog 6229* | Domain of an operation given by the maps-to notation, closed form of dmmpo 6225. 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 6230* | Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.) |
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Theorem | mpoexg 6231* | 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 6232* | 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 6233* | Weak version of mpoex 6234 that holds without ax-coll 4133. 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 6234* | 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 6235* | 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 6236* | Composition of two functions. Variation of fmptco 5699 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.) |
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Theorem | oprabco 6237* | 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 6238* | Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.) |
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Theorem | df1st2 6239* |
An alternate possible definition of the ![]() |
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Theorem | df2nd2 6240* |
An alternate possible definition of the ![]() |
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Theorem | 1stconst 6241 |
The mapping of a restriction of the ![]() |
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Theorem | 2ndconst 6242 |
The mapping of a restriction of the ![]() |
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Theorem | dfmpo 6243* |
Alternate definition for the maps-to notation df-mpo 5897 (although it
requires that ![]() |
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Theorem | cnvf1olem 6244 | Lemma for cnvf1o 6245. (Contributed by Mario Carneiro, 27-Apr-2014.) |
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Theorem | cnvf1o 6245* | 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 6246 |
The ![]() ![]() ![]() ![]() |
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Theorem | fo2ndf 6247 |
The ![]() ![]() ![]() ![]() |
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Theorem | f1o2ndf1 6248 |
The ![]() ![]() ![]() ![]() |
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Theorem | algrflem 6249 | 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 6250 | Lemma for algrf 12065 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.) |
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Theorem | xporderlem 6251* | Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.) |
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Theorem | poxp 6252* | 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 6253* | Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.) |
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Theorem | cnvoprab 6254* | The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
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Theorem | f1od2 6255* | Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
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Theorem | disjxp1 6256* | 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 6257* | 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 5897) 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 5970, ovmpox 6021 and fmpox 6220). 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 6258* |
Membership in a union of Cartesian products, using bound-variable
hypothesis for ![]() |
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Theorem | mpoxopn0yelv 6259* | 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 6260* | 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 6261* | 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 6262* | Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
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Theorem | rbropap 6263* |
Properties of a pair in a restricted binary relation ![]() ![]() ![]() ![]() |
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Syntax | ctpos 6264 | The transposition of a function. |
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Definition | df-tpos 6265* |
Define the transposition of a function, which is a function
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Theorem | tposss 6266 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposeq 6267 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposeqd 6268 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
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Theorem | tposssxp 6269 | The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
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Theorem | reltpos 6270 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | brtpos2 6271 |
Value of the transposition at a pair ![]() ![]() ![]() ![]() ![]() |
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Theorem | brtpos0 6272 | 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 6273 |
Necessary and sufficient condition for ![]() ![]() |
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Theorem | brtposg 6274 | The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.) |
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Theorem | ottposg 6275 | 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 6276 |
The domain of tpos ![]() ![]() ![]() |
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Theorem | rntpos 6277 |
The range of tpos ![]() ![]() ![]() |
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Theorem | tposexg 6278 | The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | ovtposg 6279 |
The transposition swaps the arguments in a two-argument function. When
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Theorem | tposfun 6280 | The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | dftpos2 6281* |
Alternate definition of tpos when ![]() |
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Theorem | dftpos3 6282* |
Alternate definition of tpos when ![]() |
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Theorem | dftpos4 6283* | Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tpostpos 6284 |
Value of the double transposition for a general class ![]() |
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Theorem | tpostpos2 6285 | Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
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Theorem | tposfn2 6286 | The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfo2 6287 | Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf2 6288 | The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf12 6289 | Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf1o2 6290 | Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfo 6291 | The domain and codomain/range of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf 6292 | The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfn 6293 | Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tpos0 6294 | Transposition of the empty set. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposco 6295 | Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tpossym 6296* | Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tposeqi 6297 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposex 6298 | A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | nftpos 6299 | Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposoprab 6300* | Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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