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Theorem List for Metamath Proof Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnvcnvsn 5301 Double converse of a singleton of an ordered pair. (Unlike cnvsn 5306, this does not need any sethood assumptions on and .) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnsnsn 5302 The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremrnsnopg 5303 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremrnsnop 5304 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremop1sta 5305 Extract the first member of an ordered pair. (See op2nda 5308 to extract the second member, op1stb 4712 for an alternate version, and op1st 6308 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)

Theoremcnvsn 5306 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremop2ndb 5307 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4712 to extract the first member, op2nda 5308 for an alternate version, and op2nd 6309 for the preferred version.) (Contributed by NM, 25-Nov-2003.)

Theoremop2nda 5308 Extract the second member of an ordered pair. (See op1sta 5305 to extract the first member, op2ndb 5307 for an alternate version, and op2nd 6309 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvsng 5309 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)

Theoremopswap 5310 Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.)

Theoremelxp4 5311 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5312, elxp6 6331, and elxp7 6332. (Contributed by NM, 17-Feb-2004.)

Theoremelxp5 5312 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5311 when the double intersection does not create class existence problems (caused by int0 4020). (Contributed by NM, 1-Aug-2004.)

Theoremcnvresima 5313 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)

Theoremresdm2 5314 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)

Theoremresdmres 5315 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremimadmres 5316 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremmptpreima 5317* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremmptiniseg 5318* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremdmmpt 5319 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)

Theoremdmmptss 5320* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremdmmptg 5321* The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)

Theoremrelco 5322 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)

Theoremdfco2 5323* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)

Theoremdfco2a 5324* Generalization of dfco2 5323, where can have any value between and . (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcoundi 5325 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcoundir 5326 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcores 5327 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresco 5328 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)

Theoremimaco 5329 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)

Theoremrnco 5330 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)

Theoremrnco2 5331 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)

Theoremdmco 5332 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)

Theoremcoiun 5333* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)

Theoremcocnvcnv1 5334 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)

Theoremcocnvcnv2 5335 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)

Theoremcores2 5336 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)

Theoremco02 5337 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)

Theoremco01 5338 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)

Theoremcoi1 5339 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)

Theoremcoi2 5340 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)

Theoremcoires1 5341 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)

Theoremcoass 5342 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)

Theoremrelcnvtr 5343 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)

Theoremrelssdmrn 5344 A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)

Theoremcnvssrndm 5345 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcossxp 5346 Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)

Theoremrelrelss 5347 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)

Theoremunielrel 5348 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)

Theoremrelfld 5349 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)

Theoremrelresfld 5350 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)

Theoremrelcoi2 5351 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)

Theoremrelcoi1 5352 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)

Theoremunidmrn 5353 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)

Theoremrelcnvfld 5354 if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)

Theoremdfdm2 5355 Alternate definition of domain df-dm 4842 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)

Theoremunixp 5356 The double class union of a non-empty cross product is the union of it members. (Contributed by NM, 17-Sep-2006.)

Theoremunixp0 5357 A cross product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)

Theoremunixpid 5358 Field of a square cross product. (Contributed by FL, 10-Oct-2009.)

Theoremcnvexg 5359 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)

Theoremcnvex 5360 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)

Theoremrelcnvexb 5361 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)

Theoremressn 5362 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremcnviin 5363* The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)

Theoremcnvpo 5364 The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)

Theoremcnvso 5365 The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)

Theoremcoexg 5366 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)

Theoremcoex 5367 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)

Theoremxpco 5368 Composition of two cross products. (Contributed by Thierry Arnoux, 17-Nov-2017.)

Theoremxpcoid 5369 Composition of two square cross products. (Contributed by Thierry Arnoux, 14-Jan-2018.)

2.4.8  Definite description binder (inverted iota)

Syntaxcio 5370 Extend class notation with Russell's definition description binder (inverted iota).

Theoremiotajust 5371* Soundness justification theorem for df-iota 5372. (Contributed by Andrew Salmon, 29-Jun-2011.)

Definitiondf-iota 5372* Define Russell's definition description binder, which can be read as "the unique such that ," where ordinarily contains as a free variable. Our definition is meaningful only when there is exactly one such that is true (see iotaval 5383); otherwise, it evaluates to the empty set (see iotanul 5387). Russell used the inverted iota symbol to represent the binder.

Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use riotacl2 6513 (or iotacl 5395 for unbounded iota), as demonstrated in the proof of supub 7411. This can be easier than applying riotasbc 6515 or a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF.

(Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremdfiota2 5373* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremnfiota1 5374 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfiotad 5375 Deduction version of nfiota 5376. (Contributed by NM, 18-Feb-2013.)

Theoremnfiota 5376 Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)

Theoremcbviota 5377 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Theoremcbviotav 5378* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Theoremsb8iota 5379 Variable substitution in description binder. Compare sb8eu 2270. (Contributed by NM, 18-Mar-2013.)

Theoremiotaeq 5380 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremiotabi 5381 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremuniabio 5382* Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotaval 5383* Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotauni 5384 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiotaint 5385 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremiota1 5386 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremiotanul 5387 Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one that satisfies . (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotassuni 5388 The class is a subset of the union of all elements satisfying . (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremiotaex 5389 Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiota4 5390 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiota4an 5391 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiota5 5392* A method for computing iota. (Contributed by NM, 17-Sep-2013.)

Theoremiotabidv 5393* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)

Theoremiotabii 5394 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremiotacl 5395 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5372). If you have a bounded iota-based definition, riotacl2 6513 may be useful.

(Contributed by Andrew Salmon, 1-Aug-2011.)

Theoremiota2df 5396 A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2d 5397* A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2 5398* The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremsniota 5399 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremdfiota4 5400 The operation using the operator. (Contributed by Scott Fenton, 6-Oct-2017.)

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