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Type | Label | Description |
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Statement | ||
Theorem | 1st2nd2 6201 | 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 6202 | An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.) |
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Theorem | eqop 6203 | 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 6204 | Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.) |
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Theorem | op1steq 6205* | 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 6206 | Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) |
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Theorem | 1st2nd 6207 | 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 6208 | 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 6209 | 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 6210 | 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 6211* | Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
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Theorem | reldm 6212* | An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
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Theorem | sbcopeq1a 6213 | Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2987 that avoids the existential quantifiers of copsexg 4262). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
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Theorem | csbopeq1a 6214 |
Equality theorem for substitution of a class ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dfopab2 6215* | 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 6216* | 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 6217* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.) |
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Theorem | dfoprab4 6218* | 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 6219* | 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 6220* | Define the cross product of three classes. Compare df-xp 4650. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) |
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Theorem | elopabi 6221* | 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 6222* | 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 6223* | 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 6224* | 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 6225* | 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 6226* |
Functionality, domain and codomain of a class given by the maps-to
notation, where ![]() ![]() ![]() ![]() ![]() |
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Theorem | fmpo 6227* | 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 6228* | Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
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Theorem | mpofvex 6229* | 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 6230* | Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
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Theorem | dmmpo 6231* | Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
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Theorem | mpofvexi 6232* | 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 6233* | An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.) |
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Theorem | dmmpoga 6234* | Domain of an operation given by the maps-to notation, closed form of dmmpo 6231. (Contributed by Alexander van der Vekens, 10-Feb-2019.) |
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Theorem | dmmpog 6235* | Domain of an operation given by the maps-to notation, closed form of dmmpo 6231. 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 6236* | Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.) |
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Theorem | mpoexg 6237* | 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 6238* | 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 6239* | Weak version of mpoex 6240 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 6240* | 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 6241* | 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 6242* | Composition of two functions. Variation of fmptco 5703 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.) |
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Theorem | oprabco 6243* | 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 6244* | Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.) |
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Theorem | df1st2 6245* |
An alternate possible definition of the ![]() |
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Theorem | df2nd2 6246* |
An alternate possible definition of the ![]() |
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Theorem | 1stconst 6247 |
The mapping of a restriction of the ![]() |
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Theorem | 2ndconst 6248 |
The mapping of a restriction of the ![]() |
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Theorem | dfmpo 6249* |
Alternate definition for the maps-to notation df-mpo 5902 (although it
requires that ![]() |
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Theorem | cnvf1olem 6250 | Lemma for cnvf1o 6251. (Contributed by Mario Carneiro, 27-Apr-2014.) |
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Theorem | cnvf1o 6251* | 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 6252 |
The ![]() ![]() ![]() ![]() |
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Theorem | fo2ndf 6253 |
The ![]() ![]() ![]() ![]() |
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Theorem | f1o2ndf1 6254 |
The ![]() ![]() ![]() ![]() |
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Theorem | algrflem 6255 | 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 6256 | Lemma for algrf 12080 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.) |
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Theorem | xporderlem 6257* | Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.) |
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Theorem | poxp 6258* | 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 6259* | Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.) |
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Theorem | cnvoprab 6260* | The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
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Theorem | f1od2 6261* | Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
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Theorem | disjxp1 6262* | 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 6263* | 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 5902) 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 5975, ovmpox 6026 and fmpox 6226). 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 6264* |
Membership in a union of Cartesian products, using bound-variable
hypothesis for ![]() |
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Theorem | mpoxopn0yelv 6265* | 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 6266* | 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 6267* | 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 6268* | Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
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Theorem | rbropap 6269* |
Properties of a pair in a restricted binary relation ![]() ![]() ![]() ![]() |
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Syntax | ctpos 6270 | The transposition of a function. |
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Definition | df-tpos 6271* |
Define the transposition of a function, which is a function
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Theorem | tposss 6272 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposeq 6273 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | tposeqd 6274 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
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Theorem | tposssxp 6275 | The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
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Theorem | reltpos 6276 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | brtpos2 6277 |
Value of the transposition at a pair ![]() ![]() ![]() ![]() ![]() |
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Theorem | brtpos0 6278 | 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 6279 |
Necessary and sufficient condition for ![]() ![]() |
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Theorem | brtposg 6280 | The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.) |
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Theorem | ottposg 6281 | 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 6282 |
The domain of tpos ![]() ![]() ![]() |
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Theorem | rntpos 6283 |
The range of tpos ![]() ![]() ![]() |
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Theorem | tposexg 6284 | The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | ovtposg 6285 |
The transposition swaps the arguments in a two-argument function. When
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Theorem | tposfun 6286 | The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | dftpos2 6287* |
Alternate definition of tpos when ![]() |
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Theorem | dftpos3 6288* |
Alternate definition of tpos when ![]() |
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Theorem | dftpos4 6289* | Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tpostpos 6290 |
Value of the double transposition for a general class ![]() |
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Theorem | tpostpos2 6291 | Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
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Theorem | tposfn2 6292 | The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfo2 6293 | Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf2 6294 | The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf12 6295 | Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf1o2 6296 | Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfo 6297 | The domain and codomain/range of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposf 6298 | The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) |
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Theorem | tposfn 6299 | Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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Theorem | tpos0 6300 | Transposition of the empty set. (Contributed by NM, 10-Sep-2015.) |
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