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Theorem List for Metamath Proof Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcurry2f 6401 Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)

Theoremcurry2val 6402 The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)

Theoremcnvf1olem 6403 Lemma for cnvf1o 6404. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremcnvf1o 6404* Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremfparlem1 6405 Lemma for fpar 6409. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfparlem2 6406 Lemma for fpar 6409. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfparlem3 6407* Lemma for fpar 6409. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfparlem4 6408* Lemma for fpar 6409. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfpar 6409* Merge two functions in parallel. Use as the second argument of a composition with a (2-place) operation to build compound operations such as . (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfsplit 6410 A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 6409 in order to build compound functions such as . (Contributed by NM, 17-Sep-2007.)

Theoremf2ndf 6411 The (second member of an ordered pair) function restricted to a function is a function of into the codomain of . (Contributed by Alexander van der Vekens, 4-Feb-2018.)

Theoremfo2ndf 6412 The (second member of an ordered pair) function restricted to a function is a function of onto the range of . (Contributed by Alexander van der Vekens, 4-Feb-2018.)

Theoremf1o2ndf1 6413 The (second member of an ordered pair) function restricted to a one-to-one function is a one-to-one function of onto the range of . (Contributed by Alexander van der Vekens, 4-Feb-2018.)

Theoremalgrflem 6414 Lemma for algrf 13019 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremfrxp 6415* A lexicographical ordering of two well-founded classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) (Proof shortened by Wolf Lammen, 4-Oct-2014.)

Theoremxporderlem 6416* Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)

Theorempoxp 6417* A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

Theoremsoxp 6418* A lexicographical ordering of two strictly ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

Theoremwexp 6419* A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

Theoremfnwelem 6420* Lemma for fnwe 6421. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremfnwe 6421* A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremfnse 6422* Condition for the well-order in fnwe 6421 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.)
Se               Se

2.4.14  Special "Maps to" operations

The following theorems are about maps-to operations ( see df-mpt2 6045) where 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 "mpt2x" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpt2x 6109, ovmpt2x 6161 and fmpt2x 6376). However, there is a proposal by Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2" (see beginning of set.mm). If this proposal will be realized, the labels in the following should also be adapted. If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpt2xop", and the maps-to operations are called "x-op maps-to operations" for short.

Theoremmpt2xopn0yelv 6423* 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.)

Theoremmpt2xopynvov0g 6424* If the second argument 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 is not element of the the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Theoremmpt2xopxnop0 6425* If the first argument 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, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Theoremmpt2xopx0ov0 6426* If the first argument 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, is the empty set, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Theoremmpt2xopxprcov0 6427* If the components of the first argument 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, are not sets, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Theoremmpt2xopynvov0 6428* If the second argument 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 is not element of the the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Theoremmpt2xopoveq 6429* 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.)

Theoremmpt2xopovel 6430* 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.)

Theoremmpt2xopoveqd 6431* 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, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)

Theoremmpt2ndm0 6432* The value of an operation given by a maps-to rule is the empty set if the arguments ar not contained in the base sets of the rule. (Contributed by Alexander van der Vekens, 12-Oct-2017.)

Theorembrovex 6433* A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)

Theorembrovmpt2ex 6434* A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)

Theoremsprmpt2 6435* The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)

Theoremisprmpt2 6436* Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)

2.4.15  Function transposition

Syntaxctpos 6437 The transposition of a function.
tpos

Definitiondf-tpos 6438* Define the transposition of a function, which is a function tpos satisfying . (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposss 6439 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposeq 6440 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposeqd 6441 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
tpos tpos

Theoremtposssxp 6442 The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
tpos

Theoremreltpos 6443 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos2 6444 Value of the transposition at a pair . (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos0 6445 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). This allows us to eliminate sethood hypotheses on in brtpos 6447. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremreldmtpos 6446 Necessary and sufficient condition for tpos to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos 6447 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremottpos 6448 The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
tpos

Theoremrelbrtpos 6449 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015.)
tpos

Theoremdmtpos 6450 The domain of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremrntpos 6451 The range of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposexg 6452 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremovtpos 6453 The transposition swaps the arguments in a two-argument function. When is a matrix, which is to say a function from to or some ring, tpos is the transposition of , which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposfun 6454 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremdftpos2 6455* Alternate definition of tpos when has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremdftpos3 6456* Alternate definition of tpos when has relational domain. Compare df-cnv 4845. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremdftpos4 6457* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos

Theoremtpostpos 6458 Value of the double transposition for a general class . (Contributed by Mario Carneiro, 16-Sep-2015.)
tpos tpos

Theoremtpostpos2 6459 Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
tpos tpos

Theoremtposfn2 6460 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposfo2 6461 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposf2 6462 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposf12 6463 Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposf1o2 6464 Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposfo 6465 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposf 6466 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposfn 6467 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos

Theoremtpos0 6468 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposco 6469 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos tpos

Theoremtpossym 6470* Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos

Theoremtposeqi 6471 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposex 6472 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremnftpos 6473 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposoprab 6474* Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposmpt2 6475* Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

2.4.16  Curry and uncurry

Syntaxccur 6476 Extend class notation to include the currying function.
curry

Syntaxcunc 6477 Extend class notation to include the uncurrying function.
uncurry

Definitiondf-cur 6478* Define the currying of , which splits a function of two arguments into a function of the first argument, producing a function over the second argument. (Contributed by Mario Carneiro, 7-Jan-2017.)
curry

Definitiondf-unc 6479* Define the uncurrying of , which takes a function producing functions, and transforms it into a two-argument function. (Contributed by Mario Carneiro, 7-Jan-2017.)
uncurry

2.4.17  Proper subset relation

Syntaxcrpss 6480 Extend class notation to include the reified proper subset relation.
[]

Definitiondf-rpss 6481* Define a relation which corresponds to proper subsethood df-pss 3296 on sets. This allows us to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 6486. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[]

Theoremrelrpss 6482 The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[]

Theorembrrpssg 6483 The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[]

Theorembrrpss 6484 The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[]

Theoremporpss 6485 Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[]

Theoremsorpss 6486* Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[]

Theoremsorpssi 6487 Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[]

Theoremsorpssun 6488 A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.)
[]

Theoremsorpssin 6489 A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.)
[]

Theoremsorpssuni 6490* In a chain of sets, a maximal element is the union of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[]

Theoremsorpssint 6491* In a chain of sets, a minimal element is the intersection of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[]

Theoremsorpsscmpl 6492* The componentwise complement of a chain of sets is also a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[] []

2.4.18  Iota properties

Theoremfvopab5 6493* The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremopiota 6494* The property of a uniquely specified ordered pair. (Contributed by Mario Carneiro, 21-May-2015.)

Theoremopabiotafun 6495* Define a function whose value is "the unique such that ". (Contributed by NM, 19-May-2015.)

Theoremopabiotadm 6496* Define a function whose value is "the unique such that ". (Contributed by NM, 16-Nov-2013.)

Theoremopabiota 6497* Define a function whose value is "the unique such that ". (Contributed by NM, 16-Nov-2013.)

2.4.19  Cantor's Theorem

Theoremcanth 6498 No set is equinumerous to its power set (Cantor's theorem), i.e. no function can map it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 7219. Note that must be a set: this theorem does not hold when is too large to be a set; see ncanth 6499 for a counterexample. (Use nex 1561 if you want the form .) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)

Theoremncanth 6499 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4301). Specifically, the identity function maps the universe onto its power class. Compare canth 6498 that works for sets. See also the remark in ru 3120 about NF, in which Cantor's theorem fails for sets that are "too large." This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004.)

2.4.20  Undefined values and restricted iota (description binder)

Syntaxcund 6500 Extend class notation with undefined value function.

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