Description: Define the value of a
function, , also known as function
?Error:
cos ` 0 ) = 1
^^^
This math symbol is not active (i.e. was not declared in this scope).
application. For example, (`0)=1 (we prove this in
cos012802
after we define cosine in df-cos12724). Typically, function is
defined using maps-to notation (see df-mpt4303 and df-mpt26134), but this
is not required. For example,
?Error:
2 , 6 >. , <. 3 , 9 >. } -> ( F ` 3 ) = 9
^
This math symbol is not active (i.e. was not declared in this scope).
={<.2,6>.,<.3,9>.}->(`3)=9 (ex-fv21802).
Note that df-ov6132 will 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 (as shown by ndmfv5798 and fvprc5769).
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. Alternate definitions are
dffv25844, dffv35771, fv25770,
and fv35787 (the latter two previously
required to be a set.) Restricted
equivalents that require
to be a function are shown in funfv5838 and funfv25839. For the familiar
definition of function value in terms of ordered pair membership, see
funopfvb5818. (Contributed by NM, 1-Aug-1994.) Revised
to use
iota. Original version is now theorem
dffv45772. (Revised by Scott
Fenton, 6-Oct-2017.)