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

Theoremfovcl 5801 Closure law for an operation. (Contributed by NM, 19-Apr-2007.)

Theoremeqfnov 5802* Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)

Theoremeqfnov2 5803* 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 5804* Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmpt22eqb 5805* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5803. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrnmpt2 5806* The range of an operation given by the "maps to" notation. (Contributed by FL, 20-Jun-2011.)

Theoremreldmmpt2 5807* The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Theoremelrnmpt2g 5808* Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmpt2 5809* Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremralrnmpt2 5810* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)

Theoremrexrnmpt2 5811* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)

Theoremoprabexd 5812* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremoprabex 5813* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)

Theoremoprabex3 5814* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)

Theoremoprabrexex2 5815* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)

Theoremovid 5816* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremovidig 5817* The value of an operation class abstraction. Compare ovidi 5818. The condition is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremovidi 5818* The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremov 5819* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremovigg 5820* The value of an operation class abstraction. Compare ovig 5821. The condition is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremovig 5821* 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 5822* Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5460.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremovmpt2s 5823* Value of a function given by the "maps to" notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremov2gf 5824* The value of an operation class abstraction. A version of ovmpt2g 5834 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremovmpt2dxf 5825* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremovmpt2dx 5826* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremovmpt2d 5827* Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)

Theoremovmpt2x 5828* The value of an operation class abstraction. Variant of ovmpt2ga 5829 which does not require and to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremovmpt2ga 5829* Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theoremovmpt2a 5830* Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)

Theoremovmpt2df 5831* Alternate deduction version of ovmpt2 5835, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremovmpt2dv 5832* Alternate deduction version of ovmpt2 5835, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremovmpt2dv2 5833* Alternate deduction version of ovmpt2 5835, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremovmpt2g 5834* 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 5835* 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 5836* The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremov6g 5837* The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.)

Theoremovg 5838* The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremovres 5839 The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)

Theoremovresd 5840 Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremoprssov 5841 The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.)

Theoremfovrn 5842 An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.)

Theoremfovrnda 5843 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremfovrnd 5844 An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremfnrnov 5845* The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.)

Theoremfoov 5846* An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)

Theoremfnovrn 5847 An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)

Theoremovelrn 5848* 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 5849* Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)

Theoremovelimab 5850* Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)

Theoremovconst2 5851 The value of a constant operation. (Contributed by NM, 5-Nov-2006.)

Theoremab2rexex 5852* 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 5615. (Contributed by NM, 20-Sep-2011.)

Theoremab2rexex2 5853* Existence of an existentially restricted class abstraction. is normally has free-variable parameters , , and . Compare abrexex2 5632. (Contributed by NM, 20-Sep-2011.)

Theoremoprssdm 5854* Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.)

Theoremndmovg 5855 The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.)

Theoremndmov 5856 The value of an operation outside its domain. (Contributed by NM, 24-Aug-1995.)

Theoremndmovcl 5857 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 is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by NM, 24-Sep-2004.)

Theoremndmovrcl 5858 Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.)

Theoremndmovcom 5859 Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)

Theoremndmovass 5860 Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremndmovdistr 5861 Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremndmovord 5862 Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.)

Theoremndmovordi 5863 Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.)

Theoremcaovclg 5864* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)

Theoremcaovcld 5865* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcl 5866* Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremcaovcomg 5867* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)

Theoremcaovcomd 5868* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcom 5869* Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)

Theoremcaovassg 5870* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremcaovassd 5871* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovass 5872* Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremcaovcang 5873* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcand 5874* Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcanrd 5875* Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcan 5876* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)

Theoremcaovordig 5877* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremcaovordid 5878* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremcaovordg 5879* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaovordd 5880* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovord2d 5881* Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovord3d 5882* Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovord 5883* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)

Theoremcaovord2 5884* Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)

Theoremcaovord3 5885* Ordering law. (Contributed by NM, 29-Feb-1996.)

Theoremcaovdig 5886* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)

Theoremcaovdid 5887* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovdir2d 5888* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovdirg 5889* Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)

Theoremcaovdird 5890* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovdi 5891* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)

Theoremcaov32d 5892* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov12d 5893* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov31d 5894* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov13d 5895* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov4d 5896* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov411d 5897* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov42d 5898* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov32 5899* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaov12 5900* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

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