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Type | Label | Description |
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Statement | ||
Theorem | cbviotav 4901* | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | sb8iota 4902 | Variable substitution in description binder. Compare sb8eu 1929. (Contributed by NM, 18-Mar-2013.) |
Theorem | iotaeq 4903 | Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | iotabi 4904 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | uniabio 4905* | 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 4906* | Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | iotauni 4907 | Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iotaint 4908 | Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | iota1 4909 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | iotanul 4910 | 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 4911 | 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 4912* | 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 4913 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota4an 4914 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota5 4915* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
Theorem | iotabidv 4916* | Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.) |
Theorem | iotabii 4917 | Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | iotacl 4918 |
Membership law for descriptions.
This can useful for expanding an unbounded iota-based definition (see df-iota 4895). (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | iota2df 4919 | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2d 4920* | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2 4921* | The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | sniota 4922 | A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | csbiotag 4923* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) |
Syntax | wfun 4924 | Extend the definition of a wff to include the function predicate. (Read: is a function.) |
Syntax | wfn 4925 | Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .) |
Syntax | wf 4926 | Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .) |
Syntax | wf1 4927 | 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 4928 | 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 4929 | 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 4930 | Extend the definition of a class to include the value of a function. (Read: The value of at , or " of .") |
Syntax | wiso 4931 | Extend the definition of a wff to include the isomorphism property. (Read: is an , isomorphism of onto .) |
Definition | df-fun 4932 | Define predicate that determines if some class is a function. Definition 10.1 of [Quine] p. 65. For example, the expression is true (funi 4960). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3846 with the maps-to notation (see df-mpt 3848). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4933), a function with a given domain and codomain (df-f 4934), a one-to-one function (df-f1 4935), an onto function (df-fo 4936), or a one-to-one onto function (df-f1o 4937). For alternate definitions, see dffun2 4940, dffun4 4941, dffun6 4944, dffun7 4956, dffun8 4957, and dffun9 4958. (Contributed by NM, 1-Aug-1994.) |
Definition | df-fn 4933 | Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
Definition | df-f 4934 | 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 4935 | 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 4936 | 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 4937 | 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 4938* | Define the value of a function, , also known as function application. For example, . Typically, function is defined using maps-to notation (see df-mpt 3848), 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 4939* | 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 4940* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
Theorem | dffun4 4941* | Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.) |
Theorem | dffun5r 4942* | A way of proving a relation is a function, analogous to mo2r 1968. (Contributed by Jim Kingdon, 27-May-2020.) |
Theorem | dffun6f 4943* | 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 4944* | Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) |
Theorem | funmo 4945* | A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) |
Theorem | dffun4f 4946* | Definition of function like dffun4 4941 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.) |
Theorem | funrel 4947 | A function is a relation. (Contributed by NM, 1-Aug-1994.) |
Theorem | funss 4948 | Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) |
Theorem | funeq 4949 | Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
Theorem | funeqi 4950 | Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | funeqd 4951 | Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
Theorem | nffun 4952 | Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.) |
Theorem | sbcfung 4953 | Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
Theorem | funeu 4954* | There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Theorem | funeu2 4955* | There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.) |
Theorem | dffun7 4956* | 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 4957 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.) |
Theorem | dffun8 4957* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 4956. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Theorem | dffun9 4958* | Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
Theorem | funfn 4959 | An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.) |
Theorem | funi 4960 | The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.) |
Theorem | nfunv 4961 | The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.) |
Theorem | funopg 4962 | 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 4963* | 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 4964* | A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.) |
Theorem | funopab4 4965* | A class of ordered pairs of values in the form used by df-mpt 3848 is a function. (Contributed by NM, 17-Feb-2013.) |
Theorem | funmpt 4966 | A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.) |
Theorem | funmpt2 4967 | 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 4968 | 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 4969 | A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.) |
Theorem | funssres 4970 | The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.) |
Theorem | fun2ssres 4971 | Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.) |
Theorem | funun 4972 | 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 4973 | The converse singleton of an ordered pair is a function. This is equivalent to funsn 4976 via cnvsn 4831, but stating it this way allows us to skip the sethood assumptions on and . (Contributed by NM, 30-Apr-2015.) |
Theorem | funsng 4974 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.) |
Theorem | fnsng 4975 | Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Theorem | funsn 4976 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.) |
Theorem | funprg 4977 | A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) |
Theorem | funtpg 4978 | 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 4979 | A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
Theorem | funtp 4980 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fnsn 4981 | Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | fnprg 4982 | Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | fntpg 4983 | Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
Theorem | fntp 4984 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | fun0 4985 | The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.) |
Theorem | funcnvcnv 4986 | The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.) |
Theorem | funcnv2 4987* | A simpler equivalence for single-rooted (see funcnv 4988). (Contributed by NM, 9-Aug-2004.) |
Theorem | funcnv 4988* | 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 4987 for a simpler version. (Contributed by NM, 13-Aug-2004.) |
Theorem | funcnv3 4989* | A condition showing a class is single-rooted. (See funcnv 4988). (Contributed by NM, 26-May-2006.) |
Theorem | funcnveq 4990* | Another way of expressing that a class is single-rooted. Counterpart to dffun2 4940. (Contributed by Jim Kingdon, 24-Dec-2018.) |
Theorem | fun2cnv 4991* | 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.) |
Theorem | svrelfun 4992 | A single-valued relation is a function. (See fun2cnv 4991 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.) |
Theorem | fncnv 4993* | Single-rootedness (see funcnv 4988) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.) |
Theorem | fun11 4994* | 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 "Un_{2} (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.) |
Theorem | fununi 4995* | The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.) |
Theorem | funcnvuni 4996* | The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 4988 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.) |
Theorem | fun11uni 4997* | 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.) |
Theorem | funin 4998 | 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.) |
Theorem | funres11 4999 | The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.) |
Theorem | funcnvres 5000 | The converse of a restricted function. (Contributed by NM, 27-Mar-1998.) |
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