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Theorem List for Metamath Proof Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcbvoprab3 6101* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)

Theoremcbvoprab3v 6102* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremcbvmpt2x 6103* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6104 allows to be a function of . (Contributed by NM, 29-Dec-2014.)

Theoremcbvmpt2 6104* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)

Theoremcbvmpt2v 6105* Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4254, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)

Theoremelimdelov 6106 Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). See ghomgrplem 25022 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)

Theoremdmoprab 6107* The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremdmoprabss 6108* The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)

Theoremrnoprab 6109* The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)

Theoremrnoprab2 6110* The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)

Theoremreldmoprab 6111* The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)

Theoremoprabss 6112* Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)

Theoremeloprabga 6113* The law of concretion for operation class abstraction. Compare elopab 4417. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremeloprabg 6114* The law of concretion for operation class abstraction. Compare elopab 4417. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremssoprab2i 6115* Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremmpt2v 6116* Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)

Theoremmpt2mptx 6117* Express a two-argument function as a one-argument function, or vice-versa. In this version is not assumed to be constant w.r.t . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremmpt2mpt 6118* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremresoprab 6119* Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)

Theoremresoprab2 6120* Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremresmpt2 6121* Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)

Theoremfunoprabg 6122* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)

Theoremfunoprab 6123* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.)

Theoremfnoprabg 6124* Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)

Theoremmpt2fun 6125* The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)

Theoremfnoprab 6126* Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.)

Theoremffnov 6127* An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.)

Theoremfovcl 6128 Closure law for an operation. (Contributed by NM, 19-Apr-2007.)

Theoremeqfnov 6129* Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)

Theoremeqfnov2 6130* Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.)

Theoremfnov 6131* Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmpt22eqb 6132* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 6130. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrnmpt2 6133* The range of an operation given by the "maps to" notation. (Contributed by FL, 20-Jun-2011.)

Theoremreldmmpt2 6134* The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Theoremelrnmpt2g 6135* Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmpt2 6136* Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremralrnmpt2 6137* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)

Theoremrexrnmpt2 6138* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)

Theoremoprabexd 6139* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremoprabex 6140* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)

Theoremoprabex3 6141* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)

Theoremoprabrexex2 6142* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)

Theoremovid 6143* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremovidig 6144* The value of an operation class abstraction. Compare ovidi 6145. The condition is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremovidi 6145* The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremov 6146* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremovigg 6147* The value of an operation class abstraction. Compare ovig 6148. The condition is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremovig 6148* The value of an operation class abstraction (weak version). (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremovmpt4g 6149* Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5765.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremovmpt2s 6150* Value of a function given by the "maps to" notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremov2gf 6151* The value of an operation class abstraction. A version of ovmpt2g 6161 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremovmpt2dxf 6152* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremovmpt2dx 6153* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremovmpt2d 6154* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)

Theoremovmpt2x 6155* The value of an operation class abstraction. Variant of ovmpt2ga 6156 which does not require and to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremovmpt2ga 6156* Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theoremovmpt2a 6157* Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)

Theoremovmpt2df 6158* Alternate deduction version of ovmpt2 6162, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremovmpt2dv 6159* Alternate deduction version of ovmpt2 6162, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremovmpt2dv2 6160* Alternate deduction version of ovmpt2 6162, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremovmpt2g 6161* Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremovmpt2 6162* Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremov3 6163* The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremov6g 6164* The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.)

Theoremovg 6165* The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremovres 6166 The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)

Theoremovresd 6167 Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremoprssov 6168 The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.)

Theoremfovrn 6169 An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.)

Theoremfovrnda 6170 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremfovrnd 6171 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremfnrnov 6172* The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.)

Theoremfoov 6173* An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)

Theoremfnovrn 6174 An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)

Theoremovelrn 6175* A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)

Theoremfunimassov 6176* Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)

Theoremovelimab 6177* Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)

Theoremovconst2 6178 The value of a constant operation. (Contributed by NM, 5-Nov-2006.)

Theoremab2rexex 6179* Existence of a class abstraction of existentially restricted sets. Variables and are normally free-variable parameters in the class expression substituted for , which can be thought of as . See comments for abrexex 5936. (Contributed by NM, 20-Sep-2011.)

Theoremab2rexex2 6180* Existence of an existentially restricted class abstraction. is normally has free-variable parameters , , and . Compare abrexex2 5954. (Contributed by NM, 20-Sep-2011.)

Theoremoprssdm 6181* Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.)

Theoremnssdmovg 6182 The value of an operation outside its domain. (Contributed by Alexander van der Vekens, 7-Sep-2017.)

Theoremndmovg 6183 The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.)

Theoremndmov 6184 The value of an operation outside its domain. (Contributed by NM, 24-Aug-1995.)

Theoremndmovcl 6185 The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases. It is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by NM, 24-Sep-2004.)

Theoremndmovrcl 6186 Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.)

Theoremndmovcom 6187 Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)

Theoremndmovass 6188 Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremndmovdistr 6189 Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremndmovord 6190 Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.)

Theoremndmovordi 6191 Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.)

Theoremcaovclg 6192* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)

Theoremcaovcld 6193* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcl 6194* Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremcaovcomg 6195* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)

Theoremcaovcomd 6196* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcom 6197* Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)

Theoremcaovassg 6198* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremcaovassd 6199* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovass 6200* Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)

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