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Theorem List for Intuitionistic Logic Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrelrelss 5001 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)

Theoremunielrel 5002 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)

Theoremrelfld 5003 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)

Theoremrelresfld 5004 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)

Theoremrelcoi2 5005 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)

Theoremrelcoi1 5006 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)

Theoremunidmrn 5007 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)

Theoremrelcnvfld 5008 if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)

Theoremdfdm2 5009 Alternate definition of domain df-dm 4487 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)

Theoremunixpm 5010* The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)

Theoremunixp0im 5011 The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)

Theoremcnvexg 5012 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)

Theoremcnvex 5013 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)

Theoremrelcnvexb 5014 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)

Theoremressn 5015 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremcnviinm 5016* The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)

Theoremcnvpom 5017* The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)

Theoremcnvsom 5018* The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)

Theoremcoexg 5019 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)

Theoremcoex 5020 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)

Theoremxpcom 5021* Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)

2.6.7  Definite description binder (inverted iota)

Syntaxcio 5022 Extend class notation with Russell's definition description binder (inverted iota).

Theoremiotajust 5023* Soundness justification theorem for df-iota 5024. (Contributed by Andrew Salmon, 29-Jun-2011.)

Definitiondf-iota 5024* 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 5035); otherwise, it evaluates to the empty set (see iotanul 5039). 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 iotacl 5047 (for unbounded iota). This can be easier than applying 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 5025* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremnfiota1 5026 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfiotadxy 5027* Deduction version of nfiotaxy 5028. (Contributed by Jim Kingdon, 21-Dec-2018.)

Theoremnfiotaxy 5028* Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)

Theoremcbviota 5029 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Theoremcbviotav 5030* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Theoremsb8iota 5031 Variable substitution in description binder. Compare sb8eu 1973. (Contributed by NM, 18-Mar-2013.)

Theoremiotaeq 5032 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremiotabi 5033 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremuniabio 5034* 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 5035* Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotauni 5036 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiotaint 5037 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremiota1 5038 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremiotanul 5039 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.)

Theoremeuiotaex 5040 Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)

Theoremiotass 5041* Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)

Theoremiota4 5042 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiota4an 5043 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiota5 5044* A method for computing iota. (Contributed by NM, 17-Sep-2013.)

Theoremiotabidv 5045* Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)

Theoremiotabii 5046 Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremiotacl 5047 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5024).

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

Theoremiota2df 5048 A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2d 5049* A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2 5050* The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremsniota 5051 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremcsbiotag 5052* Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)

2.6.8  Functions

Syntaxwfun 5053 Extend the definition of a wff to include the function predicate. (Read: is a function.)

Syntaxwfn 5054 Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .)

Syntaxwf 5055 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .)

Syntaxwf1 5056 Extend the definition of a wff to include one-to-one functions. (Read: maps one-to-one into .) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.

Syntaxwfo 5057 Extend the definition of a wff to include onto functions. (Read: maps onto .) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.

Syntaxwf1o 5058 Extend the definition of a wff to include one-to-one onto functions. (Read: maps one-to-one onto .) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27.

Syntaxcfv 5059 Extend the definition of a class to include the value of a function. (Read: The value of at , or " of .")

Syntaxwiso 5060 Extend the definition of a wff to include the isomorphism property. (Read: is an , isomorphism of onto .)

Definitiondf-fun 5061 Define predicate that determines if some class is a function. Definition 10.1 of [Quine] p. 65. For example, the expression is true (funi 5091). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3929 with the maps-to notation (see df-mpt 3931). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5062), a function with a given domain and codomain (df-f 5063), a one-to-one function (df-f1 5064), an onto function (df-fo 5065), or a one-to-one onto function (df-f1o 5066). For alternate definitions, see dffun2 5069, dffun4 5070, dffun6 5073, dffun7 5086, dffun8 5087, and dffun9 5088. (Contributed by NM, 1-Aug-1994.)

Definitiondf-fn 5062 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f 5063 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f1 5064 Define a one-to-one function. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow). (Contributed by NM, 1-Aug-1994.)

Definitiondf-fo 5065 Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). (Contributed by NM, 1-Aug-1994.)

Definitiondf-f1o 5066 Define a one-to-one onto function. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow). (Contributed by NM, 1-Aug-1994.)

Definitiondf-fv 5067* Define the value of a function, , also known as function application. For example, . Typically, function is defined using maps-to notation (see df-mpt 3931), but this is not required. For example, F = { 2 , 6 , 3 , 9 } -> ( F 3 ) = 9 . We will later define two-argument functions using ordered pairs as . This particular definition is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful. The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar notation for a function's value at , i.e. " of ," but without context-dependent notational ambiguity. (Contributed by NM, 1-Aug-1994.) Revised to use . (Revised by Scott Fenton, 6-Oct-2017.)

Definitiondf-isom 5068* Define the isomorphism predicate. We read this as " is an , isomorphism of onto ." Normally, and are ordering relations on and respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that and are subscripts. (Contributed by NM, 4-Mar-1997.)

Theoremdffun2 5069* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun4 5070* Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)

Theoremdffun5r 5071* A way of proving a relation is a function, analogous to mo2r 2012. (Contributed by Jim Kingdon, 27-May-2020.)

Theoremdffun6f 5072* Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremdffun6 5073* Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)

Theoremfunmo 5074* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)

Theoremdffun4f 5075* Definition of function like dffun4 5070 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.)

Theoremfunrel 5076 A function is a relation. (Contributed by NM, 1-Aug-1994.)

Theorem0nelfun 5077 A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)

Theoremfunss 5078 Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)

Theoremfuneq 5079 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)

Theoremfuneqi 5080 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremfuneqd 5081 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)

Theoremnffun 5082 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)

Theoremsbcfung 5083 Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Theoremfuneu 5084* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfuneu2 5085* There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)

Theoremdffun7 5086* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one." However, dffun8 5087 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)

Theoremdffun8 5087* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5086. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremdffun9 5088* Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Theoremfunfn 5089 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)

Theoremfunfnd 5090 A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Theoremfuni 5091 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)

Theoremnfunv 5092 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)

Theoremfunopg 5093 A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfunopab 5094* A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.)

Theoremfunopabeq 5095* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)

Theoremfunopab4 5096* A class of ordered pairs of values in the form used by df-mpt 3931 is a function. (Contributed by NM, 17-Feb-2013.)

Theoremfunmpt 5097 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremfunmpt2 5098 Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)

Theoremfunco 5099 The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfunres 5100 A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)

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