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Theorem List for Metamath Proof Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreminvdisj 4201* 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 4202* 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 4203* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsndisj 4204 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theorem0disj 4205 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjxsn 4206* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjx0 4207 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjprg 4208* A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjxiun 4209* 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 4210* The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj Disj

Theoremdisjss3 4211* 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 4212 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 9292.)

Definitiondf-br 4213 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 9125 for the specific definition of ). As a wff, relations are true or false. For example, (ex-br 21739). Often class meets the criteria to be defined in df-rel 4885, and in particular may be a function (see df-fun 5456). 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 4214 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)

Theorembreq1 4215 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)

Theorembreq2 4216 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)

Theorembreq12 4217 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqi 4218 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)

Theorembreq1i 4219 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq2i 4220 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq12i 4221 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Theorembreq1d 4222 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqd 4223 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Theorembreq2d 4224 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq12d 4225 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theorembreq123d 4226 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Theorembreqan12d 4227 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqan12rd 4228 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theoremnbrne1 4229 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)

Theoremnbrne2 4230 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)

Theoremeqbrtri 4231 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqbrtrd 4232 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)

Theoremeqbrtrri 4233 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqbrtrrd 4234 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorembreqtri 4235 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theorembreqtrd 4236 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorembreqtrri 4237 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theorembreqtrrd 4238 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorem3brtr3i 4239 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)

Theorem3brtr4i 4240 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)

Theorem3brtr3d 4241 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)

Theorem3brtr4d 4242 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)

Theorem3brtr3g 4243 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)

Theorem3brtr4g 4244 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)

Theoremsyl5eqbr 4245 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theoremsyl5eqbrr 4246 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)

Theoremsyl5breq 4247 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theoremsyl5breqr 4248 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Theoremsyl6eqbr 4249 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)

Theoremsyl6eqbrr 4250 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6breq 4251 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theoremsyl6breqr 4252 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Theoremssbrd 4253 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)

Theoremssbri 4254 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)

Theoremnfbrd 4255 Deduction version of bound-variable hypothesis builder nfbr 4256. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremnfbr 4256 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theorembrab1 4257* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)

Theorembrun 4258 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)

Theorembrin 4259 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)

Theorembrdif 4260 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)

Theoremsbcbrg 4261 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremsbcbr12g 4262* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)

Theoremsbcbr1g 4263* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)

Theoremsbcbr2g 4264* 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 4265 Extend class notation to include ordered-pair class abstraction (class builder).

Syntaxcmpt 4266 Extend the definition of a class to include maps-to notation for defining a function via a rule.

Definitiondf-opab 4267* 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 4500 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 6401. For example, (ex-opab 21740). (Contributed by NM, 4-Jul-1994.)

Definitiondf-mpt 4268* 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 4269* 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 4270 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 4271* Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)

Theoremopabbii 4272 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)

Theoremnfopab 4273* 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 4274 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 4275 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 4276* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)

Theoremcbvopabv 4277* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)

Theoremcbvopab1 4278* 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 4279* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)

Theoremcbvopab1s 4280* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)

Theoremcbvopab1v 4281* 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 4282* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)

Theoremcsbopabg 4283* Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremunopab 4284 Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Theoremmpteq12f 4285 An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq12dva 4286* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)

Theoremmpteq12dv 4287* An equality inference for the maps to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq12 4288* An equality theorem for the maps to notation. (Contributed by NM, 16-Dec-2013.)

Theoremmpteq1 4289* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq1d 4290* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)

Theoremmpteq2ia 4291 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq2i 4292 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq12i 4293 An equality inference for the maps to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq2da 4294 Slightly more general equality inference for the maps to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)

Theoremmpteq2dva 4295* Slightly more general equality inference for the maps to notation. (Contributed by Scott Fenton, 25-Apr-2012.)

Theoremmpteq2dv 4296* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)

Theoremnfmpt 4297* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)

Theoremnfmpt1 4298 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)

Theoremcbvmpt 4299* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)

Theoremcbvmptv 4300* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)

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