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

Theoremoveqd 6101 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)

Theoremoveq12d 6102 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoveqan12d 6103 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)

Theoremoveqan12rd 6104 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)

Theoremoveq123d 6105 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)

Theoremnfovd 6106 Deduction version of bound-variable hypothesis builder nfov 6107. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremnfov 6107 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)

Theoremoprabid 6108 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 20-Mar-2013.)

Theoremovex 6109 The result of an operation is a set. (Contributed by NM, 13-Mar-1995.)

Theoremovssunirn 6110 The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.)

Theoremovprc 6111 The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremovprc1 6112 The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.)

Theoremovprc2 6113 The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremovrcl 6114 Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)

Theoremcsbovg 6115 Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremcsbov12g 6116* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)

Theoremcsbov1g 6117* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)

Theoremcsbov2g 6118* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)

Theoremrspceov 6119* A frequently used special case of rspc2ev 3062 for operation values. (Contributed by NM, 21-Mar-2007.)

Theoremfnotovb 6120 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5771. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremopabbrex 6121* A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.)

Theorem0neqopab 6122 The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)

Theorembrabv 6123 If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Alexander van der Vekens, 5-Nov-2017.)

Theoremdfoprab2 6124* Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theoremreloprab 6125* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)

Theoremnfoprab1 6126 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremnfoprab2 6127 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.)

Theoremnfoprab3 6128 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)

Theoremnfoprab 6129* Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)

Theoremoprabbid 6130* Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremoprabbidv 6131* Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.)

Theoremoprabbii 6132* Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremssoprab2 6133 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4483. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Theoremssoprab2b 6134 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4484. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Theoremeqoprab2b 6135 Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4487. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremmpt2eq123 6136* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.)

Theoremmpt2eq12 6137* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpt2eq123dva 6138* An equality deduction for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)

Theoremmpt2eq123dv 6139* An equality deduction for the maps to notation. (Contributed by NM, 12-Sep-2011.)

Theoremmpt2eq123i 6140 An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.)

Theoremmpt2eq3dva 6141* Slightly more general equality inference for the maps to notation. (Contributed by NM, 17-Oct-2013.)

Theoremmpt2eq3ia 6142 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremnfmpt21 6143 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)

Theoremnfmpt22 6144 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)

Theoremnfmpt2 6145* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)

Theoremoprab4 6146* Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.)

Theoremcbvoprab1 6147* Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)

Theoremcbvoprab2 6148* Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremcbvoprab12 6149* Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremcbvoprab12v 6150* Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)

Theoremcbvoprab3 6151* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)

Theoremcbvoprab3v 6152* 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 6153* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6154 allows to be a function of . (Contributed by NM, 29-Dec-2014.)

Theoremcbvmpt2 6154* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)

Theoremcbvmpt2v 6155* Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4302, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)

Theoremelimdelov 6156 Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). See ghomgrplem 25105 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)

Theoremdmoprab 6157* The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremdmoprabss 6158* The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)

Theoremrnoprab 6159* The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)

Theoremrnoprab2 6160* The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)

Theoremreldmoprab 6161* The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)

Theoremoprabss 6162* Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)

Theoremeloprabga 6163* The law of concretion for operation class abstraction. Compare elopab 4465. (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 6164* The law of concretion for operation class abstraction. Compare elopab 4465. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremssoprab2i 6165* Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremmpt2v 6166* Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)

Theoremmpt2mptx 6167* 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 6168* 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 6169* Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)

Theoremresoprab2 6170* Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremresmpt2 6171* Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)

Theoremfunoprabg 6172* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)

Theoremfunoprab 6173* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.)

Theoremfnoprabg 6174* Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)

Theoremmpt2fun 6175* The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)

Theoremfnoprab 6176* Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.)

Theoremffnov 6177* An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.)

Theoremfovcl 6178 Closure law for an operation. (Contributed by NM, 19-Apr-2007.)

Theoremeqfnov 6179* Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)

Theoremeqfnov2 6180* 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 6181* Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmpt22eqb 6182* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 6180. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrnmpt2 6183* The range of an operation given by the "maps to" notation. (Contributed by FL, 20-Jun-2011.)

Theoremreldmmpt2 6184* The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Theoremelrnmpt2g 6185* Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmpt2 6186* Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremralrnmpt2 6187* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)

Theoremrexrnmpt2 6188* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)

Theoremoprabexd 6189* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremoprabex 6190* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)

Theoremoprabex3 6191* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)

Theoremoprabrexex2 6192* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)

Theoremovid 6193* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremovidig 6194* The value of an operation class abstraction. Compare ovidi 6195. The condition is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremovidi 6195* The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremov 6196* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremovigg 6197* The value of an operation class abstraction. Compare ovig 6198. The condition is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremovig 6198* 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 6199* Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5815.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremovmpt2s 6200* Value of a function given by the "maps to" notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)

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