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Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcbvdisjv 3901* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
Disj Disj

Theoremnfdisj 3902 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremnfdisj1 3903 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjor 3904* Two ways to say that a collection for is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
Disj

TheoremdisjmoOLD 3905* Two ways to say that a collection for is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.)

Theoremdisjors 3906* Two ways to say that a collection for is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisji2 3907* Property of a disjoint collection: if and , and , then and are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisji 3908* Property of a disjoint collection: if and have a common element , then . (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoreminvdisj 3909* If there is a function such that for all , then the sets for distinct are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
Disj

Theoremdisjiun 3910* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
Disj

TheoremdisjiunOLD 3911* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.)

Theoremsndisj 3912 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theorem0disj 3913 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjxsn 3914* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjx0 3915 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjprg 3916* A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjxiun 3917* An indexed union of a disjoint collection of disjoint collections is disjoint if each component is disjoint, and the disjoint unions in the collection are also disjoint. Note that and may have the displayed free variables. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj Disj Disj

Theoremdisjxun 3918* The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj Disj

Theoremdisjss3 3919* Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
Disj Disj

2.1.22  Binary relations

Syntaxwbr 3920 Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 8918.)

Definitiondf-br 3921 Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. Class often denotes a relation such as " " that compares two classes and , which might be numbers such as and (see df-ltxr 8752 for the specific definition of ). As a wff, relations are true or false. For example, (ex-br 20631). Often class meets the criteria to be defined in df-rel 4595, and in particular may be a function (see df-fun 4602). This definition of relations is well-defined, although not very meaningful, when classes and/or are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when is a proper class. (Contributed by NM, 31-Dec-1993.)

Theorembreq 3922 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)

Theorembreq1 3923 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)

Theorembreq2 3924 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)

Theorembreq12 3925 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqi 3926 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)

Theorembreq1i 3927 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq2i 3928 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq12i 3929 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Theorembreq1d 3930 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqd 3931 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Theorembreq2d 3932 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq12d 3933 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theorembreq123d 3934 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Theorembreqan12d 3935 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqan12rd 3936 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theoremnbrne1 3937 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)

Theoremnbrne2 3938 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)

Theoremeqbrtri 3939 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqbrtrd 3940 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)

Theoremeqbrtrri 3941 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqbrtrrd 3942 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorembreqtri 3943 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theorembreqtrd 3944 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorembreqtrri 3945 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theorembreqtrrd 3946 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorem3brtr3i 3947 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)

Theorem3brtr4i 3948 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)

Theorem3brtr3d 3949 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)

Theorem3brtr4d 3950 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)

Theorem3brtr3g 3951 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)

Theorem3brtr4g 3952 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)

Theoremsyl5eqbr 3953 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theoremsyl5eqbrr 3954 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)

Theoremsyl5breq 3955 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theoremsyl5breqr 3956 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Theoremsyl6eqbr 3957 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)

Theoremsyl6eqbrr 3958 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6breq 3959 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theoremsyl6breqr 3960 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Theoremssbrd 3961 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)

Theoremssbri 3962 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)

Theoremnfbrd 3963 Deduction version of bound-variable hypothesis builder nfbr 3964. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremnfbr 3964 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theorembrab1 3965* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)

Theorembrun 3966 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)

Theorembrin 3967 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)

Theorembrdif 3968 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)

Theoremsbcbrg 3969 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremsbcbr12g 3970* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)

Theoremsbcbr1g 3971* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)

Theoremsbcbr2g 3972* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)

2.1.23  Ordered-pair class abstractions (class builders)

Syntaxcopab 3973 Extend class notation to include ordered-pair class abstraction (class builder).

Syntaxcmpt 3974 Extend the definition of a class to include maps-to notation for defining a function via a rule.

Definitiondf-opab 3975* Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually and are distinct, although the definition doesn't strictly require it (see dfid2 4204 for a case where they are not distinct). The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. An alternate definition using no existential quantifiers is shown by dfopab2 6026. For example, (ex-opab 20632). (Contributed by NM, 4-Jul-1994.)

Definitiondf-mpt 3976* Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from (in ) to ." The class expression is the value of the function at and normally contains the variable . An example is the square function for complex numbers, . Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)

Theoremopabss 3977* The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremopabbid 3978 Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremopabbidv 3979* Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)

Theoremopabbii 3980 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)

Theoremnfopab 3981* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)

Theoremnfopab1 3982 The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremnfopab2 3983 The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcbvopab 3984* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)

Theoremcbvopabv 3985* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)

Theoremcbvopab1 3986* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcbvopab2 3987* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)

Theoremcbvopab1s 3988* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)

Theoremcbvopab1v 3989* Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Theoremcbvopab2v 3990* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)

Theoremcsbopabg 3991* Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremunopab 3992 Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Theoremmpteq12f 3993 An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq12dva 3994* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)

Theoremmpteq12dv 3995* An equality inference for the maps to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq12 3996* An equality theorem for the maps to notation. (Contributed by NM, 16-Dec-2013.)

Theoremmpteq1 3997* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq1d 3998* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremmpteq2ia 3999 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq2i 4000 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

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