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Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmpt2mpt 6201* 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 6202* Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)

Theoremresoprab2 6203* Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremresmpt2 6204* Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)

Theoremfunoprabg 6205* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)

Theoremfunoprab 6206* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.)

Theoremfnoprabg 6207* Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)

Theoremmpt2fun 6208* The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)

Theoremfnoprab 6209* Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.)

Theoremffnov 6210* An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.)

Theoremfovcl 6211 Closure law for an operation. (Contributed by NM, 19-Apr-2007.)

Theoremeqfnov 6212* Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)

Theoremeqfnov2 6213* 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 6214* Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmpt22eqb 6215* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 6213. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrnmpt2 6216* The range of an operation given by the "maps to" notation. (Contributed by FL, 20-Jun-2011.)

Theoremreldmmpt2 6217* The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Theoremelrnmpt2g 6218* Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmpt2 6219* Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremralrnmpt2 6220* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)

Theoremrexrnmpt2 6221* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)

Theoremoprabexd 6222* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremoprabex 6223* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)

Theoremoprabex3 6224* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)

Theoremoprabrexex2 6225* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)

Theoremovid 6226* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremovidig 6227* The value of an operation class abstraction. Compare ovidi 6228. The condition is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremovidi 6228* The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremov 6229* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremovigg 6230* The value of an operation class abstraction. Compare ovig 6231. The condition is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremovig 6231* 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 6232* Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5848.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremovmpt2s 6233* Value of a function given by the "maps to" notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremov2gf 6234* The value of an operation class abstraction. A version of ovmpt2g 6244 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremovmpt2dxf 6235* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremovmpt2dx 6236* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremovmpt2d 6237* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)

Theoremovmpt2x 6238* The value of an operation class abstraction. Variant of ovmpt2ga 6239 which does not require and to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremovmpt2ga 6239* Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theoremovmpt2a 6240* Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)

Theoremovmpt2df 6241* Alternate deduction version of ovmpt2 6245, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremovmpt2dv 6242* Alternate deduction version of ovmpt2 6245, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremovmpt2dv2 6243* Alternate deduction version of ovmpt2 6245, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremovmpt2g 6244* 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 6245* 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 6246* The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremov6g 6247* The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.)

Theoremovg 6248* The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremovres 6249 The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)

Theoremovresd 6250 Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremoprssov 6251 The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.)

Theoremfovrn 6252 An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.)

Theoremfovrnda 6253 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremfovrnd 6254 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremfnrnov 6255* The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.)

Theoremfoov 6256* An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)

Theoremfnovrn 6257 An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)

Theoremovelrn 6258* 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 6259* Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)

Theoremovelimab 6260* Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)

Theoremovconst2 6261 The value of a constant operation. (Contributed by NM, 5-Nov-2006.)

Theoremab2rexex 6262* 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 6019. (Contributed by NM, 20-Sep-2011.)

Theoremab2rexex2 6263* Existence of an existentially restricted class abstraction. is normally has free-variable parameters , , and . Compare abrexex2 6037. (Contributed by NM, 20-Sep-2011.)

Theoremoprssdm 6264* Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.)

Theoremnssdmovg 6265 The value of an operation outside its domain. (Contributed by Alexander van der Vekens, 7-Sep-2017.)

Theoremndmovg 6266 The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.)

Theoremndmov 6267 The value of an operation outside its domain. (Contributed by NM, 24-Aug-1995.)

Theoremndmovcl 6268 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 6269 Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.)

Theoremndmovcom 6270 Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)

Theoremndmovass 6271 Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremndmovdistr 6272 Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremndmovord 6273 Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.)

Theoremndmovordi 6274 Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.)

Theoremcaovclg 6275* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)

Theoremcaovcld 6276* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcl 6277* Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremcaovcomg 6278* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)

Theoremcaovcomd 6279* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcom 6280* Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)

Theoremcaovassg 6281* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremcaovassd 6282* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovass 6283* Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremcaovcang 6284* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcand 6285* Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcanrd 6286* Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcan 6287* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)

Theoremcaovordig 6288* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremcaovordid 6289* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremcaovordg 6290* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaovordd 6291* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovord2d 6292* Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovord3d 6293* Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovord 6294* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)

Theoremcaovord2 6295* Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)

Theoremcaovord3 6296* Ordering law. (Contributed by NM, 29-Feb-1996.)

Theoremcaovdig 6297* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)

Theoremcaovdid 6298* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovdir2d 6299* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovdirg 6300* Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)

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