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Theorem List for Intuitionistic Logic Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem1st2nd2 5801 Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)

Theoremxpopth 5802 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)

Theoremeqop 5803 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)

Theoremeqop2 5804 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)

Theoremop1steq 5805* 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.)

Theorem2nd1st 5806 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)

Theorem1st2nd 5807 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)

Theorem1stdm 5808 The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)

Theorem2ndrn 5809 The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)

Theorem1st2ndbr 5810 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremreleldm2 5811* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Theoremreldm 5812* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Theoremsbcopeq1a 5813 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2773 that avoids the existential quantifiers of copsexg 3981). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremcsbopeq1a 5814 Equality theorem for substitution of a class for an ordered pair in (analog of csbeq1a 2860). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfopab2 5815* 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.)

Theoremdfoprab3s 5816* 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.)

Theoremdfoprab3 5817* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)

Theoremdfoprab4 5818* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfoprab4f 5819* 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.)

Theoremdfxp3 5820* Define the cross product of three classes. Compare df-xp 4351. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)

Theoremelopabi 5821* A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)

Theoremeloprabi 5822* 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.)

Theoremmpt2mptsx 5823* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremmpt2mpts 5824* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)

Theoremdmmpt2ssx 5825* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremfmpt2x 5826* 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.)

Theoremfmpt2 5827* Functionality, domain and range of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremfnmpt2 5828* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremmpt2fvex 5829* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Theoremfnmpt2i 5830* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremdmmpt2 5831* Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremmpt2fvexi 5832* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Theoremmpt2exxg 5833* Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremmpt2exg 5834* Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremmpt2exga 5835* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 12-Sep-2011.)

Theoremmpt2ex 5836* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)

Theoremfmpt2co 5837* Composition of two functions. Variation of fmptco 5330 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theoremoprabco 5838* Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Theoremoprab2co 5839* Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)

Theoremdf1st2 5840* An alternate possible definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdf2nd2 5841* An alternate possible definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theorem1stconst 5842 The mapping of a restriction of the function to a constant function. (Contributed by NM, 14-Dec-2008.)

Theorem2ndconst 5843 The mapping of a restriction of the function to a converse constant function. (Contributed by NM, 27-Mar-2008.)

Theoremdfmpt2 5844* Alternate definition for the "maps to" notation df-mpt2 5517 (although it requires that be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremcnvf1olem 5845 Lemma for cnvf1o 5846. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremcnvf1o 5846* Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremf2ndf 5847 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 5848 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 5849 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 5850 Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremalgrflemg 5851 Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.)

Theoremxporderlem 5852* Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)

Theorempoxp 5853* A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

2.6.15  Special "Maps to" operations

The following theorems are about maps-to operations (see df-mpt2 5517) 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 5582, ovmpt2x 5629 and fmpt2x 5826). 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 5854* 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.)

Theoremmpt2xopoveq 5855* 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 5856* 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.)

Theoremsprmpt2 5857* 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 5858* Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)

2.6.16  Function transposition

Syntaxctpos 5859 The transposition of a function.
tpos

Definitiondf-tpos 5860* Define the transposition of a function, which is a function tpos satisfying . (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposss 5861 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposeq 5862 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposeqd 5863 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
tpos tpos

Theoremtposssxp 5864 The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
tpos

Theoremreltpos 5865 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos2 5866 Value of the transposition at a pair . (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos0 5867 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.)
tpos

Theoremreldmtpos 5868 Necessary and sufficient condition for tpos to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtposg 5869 The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.)
tpos

Theoremottposg 5870 The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
tpos

Theoremdmtpos 5871 The domain of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremrntpos 5872 The range of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposexg 5873 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremovtposg 5874 The transposition swaps the arguments in a two-argument function. When is a matrix, which is to say a function from ( 1 ... m ) ( 1 ... n ) to the reals or some ring, tpos is the transposition of , which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposfun 5875 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremdftpos2 5876* Alternate definition of tpos when has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremdftpos3 5877* Alternate definition of tpos when has relational domain. Compare df-cnv 4353. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremdftpos4 5878* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos

Theoremtpostpos 5879 Value of the double transposition for a general class . (Contributed by Mario Carneiro, 16-Sep-2015.)
tpos tpos

Theoremtpostpos2 5880 Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
tpos tpos

Theoremtposfn2 5881 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposfo2 5882 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposf2 5883 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposf12 5884 Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposf1o2 5885 Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposfo 5886 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposf 5887 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposfn 5888 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos

Theoremtpos0 5889 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
tpos

Theoremtposco 5890 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos tpos

Theoremtpossym 5891* Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos

Theoremtposeqi 5892 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposex 5893 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremnftpos 5894 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposoprab 5895* Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposmpt2 5896* Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

2.6.17  Undefined values

Theorempwuninel2 5897 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.)

Theorem2pwuninelg 5898 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.)

2.6.18  Functions on ordinals; strictly monotone ordinal functions

Theoremiunon 5899* The indexed union of a set of ordinal numbers is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)

Syntaxwsmo 5900 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.

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