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Theorem List for Metamath Proof Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxp1st 6001 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremxp2nd 6002 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremelxp6 6003 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5066. (Contributed by NM, 9-Oct-2004.)

Theoremelxp7 6004 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5066. (Contributed by NM, 19-Aug-2006.)

Theoremdifxp 6005 Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.)

Theoremdifxp1 6006 Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)

Theoremdifxp2 6007 Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)

Theoremeqopi 6008 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)

Theoremxp2 6009* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)

Theoremunielxp 6010 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)

Theorem1st2nd2 6011 Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)

Theorem1st2ndb 6012 Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.)

Theoremxpopth 6013 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)

Theoremeqop 6014 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)

Theoremeqop2 6015 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)

Theoremop1steq 6016* 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 6017 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)

Theorem1st2nd 6018 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)

Theorem1stdm 6019 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 6020 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 6021 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremreleldm2 6022* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Theoremreldm 6023* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Theoremsbcopeq1a 6024 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2931 that avoids the existential quantifiers of copsexg 4145). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremcsbopeq1a 6025 Equality theorem for substitution of a class for an ordered pair in (analog of csbeq1a 3017). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfopab2 6026* 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 6027* 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 6028* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)

Theoremdfoprab4 6029* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfoprab4f 6030* 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 6031* Define the cross product of three classes. Compare df-xp 4594. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)

Theoremcopsex2gb 6032* Implicit substitution inference for ordered pairs. Compare copsex2ga 6033. (Contributed by NM, 12-Mar-2014.)

Theoremcopsex2ga 6033* Implicit substitution inference for ordered pairs. Compare copsex2g 4147. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremelopaba 6034* Membership in an ordered pair class builder. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremexopxfr 6035* Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremexopxfr2 6036* Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.)

Theoremelopabi 6037* A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)

Theoremeloprabi 6038* 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 6039* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremmpt2mpts 6040* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)

Theoremdmmpt2ssx 6041* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremfmpt2x 6042* Functionality, domain and range of a class given by the "maps to" notation, where is not constant but depends on . (Contributed by NM, 29-Dec-2014.)

Theoremfmpt2 6043* Functionality, domain and range of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremfnmpt2 6044* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremfnmpt2i 6045* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremdmmpt2 6046* Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremmpt2exxg 6047* Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremmpt2exg 6048* Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremmpt2exga 6049* 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 6050* 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.)

Theoremmpt20 6051 A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)

Theoremovmptss 6052* If all the values of the mapping are subsets of a class , then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremrelmpt2opab 6053* Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremfmpt2co 6054* Composition of two functions. Variation of fmptco 5543 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theoremoprabco 6055* Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Theoremoprab2co 6056* Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)

Theoremdf1st2 6057* An alternate possible definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdf2nd2 6058* An alternate possible definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theorem1stconst 6059 The mapping of a restriction of the function to a constant function. (Contributed by NM, 14-Dec-2008.)

Theorem2ndconst 6060 The mapping of a restriction of the function to a converse constant function. (Contributed by NM, 27-Mar-2008.)

Theoremdfmpt2 6061* Alternate definition for the "maps to" notation df-mpt2 5715 (although it requires that be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremcurry1 6062* Composition with turns any binary operation with a constant first operand into a function of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremcurry1val 6063 The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremcurry1f 6064 Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)

Theoremcurry2 6065* Composition with turns any binary operation with a constant second operand into a function of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.)

Theoremcurry2f 6066 Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)

Theoremcurry2val 6067 The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)

Theoremcnvf1olem 6068 Lemma for cnvf1o 6069. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremcnvf1o 6069* Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremfparlem1 6070 Lemma for fpar 6074. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfparlem2 6071 Lemma for fpar 6074. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfparlem3 6072* Lemma for fpar 6074. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfparlem4 6073* Lemma for fpar 6074. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfpar 6074* 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 6075 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 6074 in order to build compound functions such as . (Contributed by NM, 17-Sep-2007.)

Theoremalgrflem 6076 Lemma for algrf 12617 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremfrxp 6077* 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 6078* Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)

Theorempoxp 6079* A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

Theoremsoxp 6080* A lexicographical ordering of two strictly ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

Theoremwexp 6081* A lexicographical ordering of two well ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

Theoremfnwelem 6082* Lemma for fnwe 6083. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremfnwe 6083* 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 6084* Condition for the well-order in fnwe 6083 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.)
Se               Se

2.4.12  Function transposition

Syntaxctpos 6085 The transposition of a function.
tpos

Definitiondf-tpos 6086* Define the transposition of a function, which is a function tpos satisfying . (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposss 6087 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposeq 6088 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposeqd 6089 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
tpos tpos

Theoremtposssxp 6090 The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
tpos

Theoremreltpos 6091 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos2 6092 Value of the transposition at a pair . (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos0 6093 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 6095. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremreldmtpos 6094 Necessary and sufficient condition for tpos to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos 6095 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremottpos 6096 The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
tpos

Theoremrelbrtpos 6097 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015.)
tpos

Theoremdmtpos 6098 The domain of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremrntpos 6099 The range of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposexg 6100 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

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