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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ressn 4801 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Theorem | cnviinm 4802* | The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvpom 4803* | The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.) |
Theorem | cnvsom 4804* | The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Theorem | coexg 4805 | The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
Theorem | coex 4806 | The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.) |
Theorem | xpcom 4807* | Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.) |
Syntax | cio 4808 | Extend class notation with Russell's definition description binder (inverted iota). |
Theorem | iotajust 4809* | Soundness justification theorem for df-iota 4810. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Definition | df-iota 4810* |
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 4821);
otherwise, it evaluates to the empty set (see iotanul 4825). 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 4833 (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.) |
Theorem | dfiota2 4811* | Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | nfiota1 4812 | Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | nfiotadxy 4813* | Deduction version of nfiotaxy 4814. (Contributed by Jim Kingdon, 21-Dec-2018.) |
Theorem | nfiotaxy 4814* | Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.) |
Theorem | cbviota 4815 | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | cbviotav 4816* | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | sb8iota 4817 | Variable substitution in description binder. Compare sb8eu 1910. (Contributed by NM, 18-Mar-2013.) |
Theorem | iotaeq 4818 | Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | iotabi 4819 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | uniabio 4820* | 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.) |
Theorem | iotaval 4821* | Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | iotauni 4822 | Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iotaint 4823 | Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | iota1 4824 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | iotanul 4825 | 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.) |
Theorem | euiotaex 4826 | 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.) |
Theorem | iotass 4827* | 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.) |
Theorem | iota4 4828 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota4an 4829 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota5 4830* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
Theorem | iotabidv 4831* | Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.) |
Theorem | iotabii 4832 | Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | iotacl 4833 |
Membership law for descriptions.
This can useful for expanding an unbounded iota-based definition (see df-iota 4810). (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | iota2df 4834 | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2d 4835* | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2 4836* | The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | sniota 4837 | A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | csbiotag 4838* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) |
Syntax | wfun 4839 | Extend the definition of a wff to include the function predicate. (Read: is a function.) |
Syntax | wfn 4840 | Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .) |
Syntax | wf 4841 | Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .) |
Syntax | wf1 4842 | 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. |
Syntax | wfo 4843 | 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. |
Syntax | wf1o 4844 | 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. |
Syntax | cfv 4845 | Extend the definition of a class to include the value of a function. (Read: The value of at , or " of .") |
Syntax | wiso 4846 | Extend the definition of a wff to include the isomorphism property. (Read: is an , isomorphism of onto .) |
Definition | df-fun 4847 | Define predicate that determines if some class is a function. Definition 10.1 of [Quine] p. 65. For example, the expression is true (funi 4875). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3809 with the maps-to notation (see df-mpt 3811). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4848), a function with a given domain and codomain (df-f 4849), a one-to-one function (df-f1 4850), an onto function (df-fo 4851), or a one-to-one onto function (df-f1o 4852). For alternate definitions, see dffun2 4855, dffun4 4856, dffun6 4859, dffun7 4871, dffun8 4872, and dffun9 4873. (Contributed by NM, 1-Aug-1994.) |
Definition | df-fn 4848 | Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
Definition | df-f 4849 | Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
Definition | df-f1 4850 | 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.) |
Definition | df-fo 4851 | 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.) |
Definition | df-f1o 4852 | 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.) |
Definition | df-fv 4853* | Define the value of a function, , also known as function application. For example, . Typically, function is defined using maps-to notation (see df-mpt 3811), 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.) |
Definition | df-isom 4854* | 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.) |
Theorem | dffun2 4855* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
Theorem | dffun4 4856* | Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.) |
Theorem | dffun5r 4857* | A way of proving a relation is a function, analogous to mo2r 1949. (Contributed by Jim Kingdon, 27-May-2020.) |
Theorem | dffun6f 4858* | 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.) |
Theorem | dffun6 4859* | Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) |
Theorem | funmo 4860* | A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) |
Theorem | dffun4f 4861* | Definition of function like dffun4 4856 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.) |
Theorem | funrel 4862 | A function is a relation. (Contributed by NM, 1-Aug-1994.) |
Theorem | funss 4863 | Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) |
Theorem | funeq 4864 | Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
Theorem | funeqi 4865 | Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | funeqd 4866 | Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
Theorem | nffun 4867 | Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.) |
Theorem | sbcfung 4868 | Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
Theorem | funeu 4869* | There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Theorem | funeu2 4870* | There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.) |
Theorem | dffun7 4871* | 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 4872 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.) |
Theorem | dffun8 4872* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 4871. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Theorem | dffun9 4873* | Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
Theorem | funfn 4874 | An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.) |
Theorem | funi 4875 | The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.) |
Theorem | nfunv 4876 | The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.) |
Theorem | funopg 4877 | 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.) |
Theorem | funopab 4878* | 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.) |
Theorem | funopabeq 4879* | A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.) |
Theorem | funopab4 4880* | A class of ordered pairs of values in the form used by df-mpt 3811 is a function. (Contributed by NM, 17-Feb-2013.) |
Theorem | funmpt 4881 | A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.) |
Theorem | funmpt2 4882 | Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.) |
Theorem | funco 4883 | 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.) |
Theorem | funres 4884 | A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.) |
Theorem | funssres 4885 | The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.) |
Theorem | fun2ssres 4886 | Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.) |
Theorem | funun 4887 | The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.) |
Theorem | funcnvsn 4888 | The converse singleton of an ordered pair is a function. This is equivalent to funsn 4891 via cnvsn 4746, but stating it this way allows us to skip the sethood assumptions on and . (Contributed by NM, 30-Apr-2015.) |
Theorem | funsng 4889 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.) |
Theorem | fnsng 4890 | Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Theorem | funsn 4891 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.) |
Theorem | funprg 4892 | A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) |
Theorem | funtpg 4893 | A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
Theorem | funpr 4894 | A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
Theorem | funtp 4895 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fnsn 4896 | Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | fnprg 4897 | Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | fntpg 4898 | Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
Theorem | fntp 4899 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | fun0 4900 | The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.) |
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