**Description: **Define the value of a
function, ,
also known as function
application. For example, (we prove this in cos0 12425
after we define cosine in df-cos 12347). Typically function is
defined using maps-to notation (see df-mpt 4079 and df-mpt2 5824), but this
is not required. For example,
(ex-fv 20806).
Note that df-ov 5822 will define two-argument functions using
ordered pairs
as .
Although based on the idea
embodied by Definition 10.2 of [Quine] p.
65 (see args 5039), our
definition apparently does not appear in the literature. However, it is
quite convenient: it can be applied to any class and evaluates to the
empty set when it is not meaningful (as shown by ndmfv 5513 and fvprc 5482).
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
dffv2 5553 and dffv3 6280. For other alternate definitions
(that fail to
evaluate to the empty set for proper classes), see fv2 5481,
fv3 5501, and
fv4 6281. Restricted equivalents that require to be a function are
shown in funfv 5547 and funfv2 5548. For the familiar definition of function
value in terms of ordered pair membership, see funopfvb 5527.
(Contributed by NM, 1-Aug-1994.) |