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Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsb8iota 5101 Variable substitution in description binder. Compare sb8eu 2013. (Contributed by NM, 18-Mar-2013.)

Theoremiotaeq 5102 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremiotabi 5103 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremuniabio 5104* 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 5105* Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotauni 5106 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiotaint 5107 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremiota1 5108 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremiotanul 5109 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 5110 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 5111* 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 5112 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiota4an 5113 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiota5 5114* A method for computing iota. (Contributed by NM, 17-Sep-2013.)

Theoremiotabidv 5115* Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)

Theoremiotabii 5116 Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremiotacl 5117 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5094).

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

Theoremiota2df 5118 A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2d 5119* A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2 5120* The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremsniota 5121 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremcsbiotag 5122* Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)

2.6.8  Functions

Syntaxwfun 5123 Extend the definition of a wff to include the function predicate. (Read: is a function.)

Syntaxwfn 5124 Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .)

Syntaxwf 5125 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .)

Syntaxwf1 5126 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 5127 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 5128 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 5129 Extend the definition of a class to include the value of a function. (Read: The value of at , or " of .")

Syntaxwiso 5130 Extend the definition of a wff to include the isomorphism property. (Read: is an , isomorphism of onto .)

Definitiondf-fun 5131 Define predicate that determines if some class is a function. Definition 10.1 of [Quine] p. 65. For example, the expression is true (funi 5161). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3995 with the maps-to notation (see df-mpt 3997). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5132), a function with a given domain and codomain (df-f 5133), a one-to-one function (df-f1 5134), an onto function (df-fo 5135), or a one-to-one onto function (df-f1o 5136). For alternate definitions, see dffun2 5139, dffun4 5140, dffun6 5143, dffun7 5156, dffun8 5157, and dffun9 5158. (Contributed by NM, 1-Aug-1994.)

Definitiondf-fn 5132 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f 5133 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)

Definitiondf-f1 5134 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 5135 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 5136 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 5137* Define the value of a function, , also known as function application. For example, . Typically, function is defined using maps-to notation (see df-mpt 3997), 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 5138* 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 5139* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun4 5140* Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)

Theoremdffun5r 5141* A way of proving a relation is a function, analogous to mo2r 2052. (Contributed by Jim Kingdon, 27-May-2020.)

Theoremdffun6f 5142* 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 5143* Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)

Theoremfunmo 5144* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)

Theoremdffun4f 5145* Definition of function like dffun4 5140 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.)

Theoremfunrel 5146 A function is a relation. (Contributed by NM, 1-Aug-1994.)

Theorem0nelfun 5147 A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)

Theoremfunss 5148 Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)

Theoremfuneq 5149 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)

Theoremfuneqi 5150 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremfuneqd 5151 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)

Theoremnffun 5152 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)

Theoremsbcfung 5153 Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Theoremfuneu 5154* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfuneu2 5155* There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)

Theoremdffun7 5156* 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 5157 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)

Theoremdffun8 5157* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5156. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremdffun9 5158* Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Theoremfunfn 5159 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)

Theoremfunfnd 5160 A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Theoremfuni 5161 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)

Theoremnfunv 5162 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)

Theoremfunopg 5163 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 5164* 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 5165* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)

Theoremfunopab4 5166* A class of ordered pairs of values in the form used by df-mpt 3997 is a function. (Contributed by NM, 17-Feb-2013.)

Theoremfunmpt 5167 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremfunmpt2 5168 Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)

Theoremfunco 5169 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 5170 A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)

Theoremfunssres 5171 The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)

Theoremfun2ssres 5172 Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)

Theoremfunun 5173 The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.)

Theoremfuncnvsn 5174 The converse singleton of an ordered pair is a function. This is equivalent to funsn 5177 via cnvsn 5027, but stating it this way allows us to skip the sethood assumptions on and . (Contributed by NM, 30-Apr-2015.)

Theoremfunsng 5175 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)

Theoremfnsng 5176 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremfunsn 5177 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)

Theoremfuninsn 5178 A function based on the singleton of an ordered pair. Unlike funsng 5175, this holds even if or is a proper class. (Contributed by Jim Kingdon, 17-Apr-2022.)

Theoremfunprg 5179 A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)

Theoremfuntpg 5180 A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.)

Theoremfunpr 5181 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfuntp 5182 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfnsn 5183 Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremfnprg 5184 Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfntpg 5185 Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)

Theoremfntp 5186 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfun0 5187 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)

Theoremfuncnvcnv 5188 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)

Theoremfuncnv2 5189* A simpler equivalence for single-rooted (see funcnv 5190). (Contributed by NM, 9-Aug-2004.)

Theoremfuncnv 5190* The converse of a class is a function iff the class is single-rooted, which means that for any in the range of there is at most one such that . Definition of single-rooted in [Enderton] p. 43. See funcnv2 5189 for a simpler version. (Contributed by NM, 13-Aug-2004.)

Theoremfuncnv3 5191* A condition showing a class is single-rooted. (See funcnv 5190). (Contributed by NM, 26-May-2006.)

Theoremfuncnveq 5192* Another way of expressing that a class is single-rooted. Counterpart to dffun2 5139. (Contributed by Jim Kingdon, 24-Dec-2018.)

Theoremfun2cnv 5193* The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that is not necessarily a function. (Contributed by NM, 13-Aug-2004.)

Theoremsvrelfun 5194 A single-valued relation is a function. (See fun2cnv 5193 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)

Theoremfncnv 5195* Single-rootedness (see funcnv 5190) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)

Theoremfun11 5196* Two ways of stating that is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)

Theoremfununi 5197* The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)

Theoremfuncnvuni 5198* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5190 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)

Theoremfun11uni 5199* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)

Theoremfunin 5200 The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

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