| Description: Define the value of a
function, (𝐹‘𝐴), also known as function
application. For example, (cos‘0) = 1 (we
prove this in cos0 14880
after we define cosine in df-cos 14801). Typically, function 𝐹 is
defined using maps-to notation (see df-mpt 4730 and df-mpt2 6655), but this
is not required. For example,
𝐹 =
{⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 27300).
Note that df-ov 6653 will define two-argument functions using
ordered pairs
as (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩). 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 ndmfv 6218 and fvprc 6185).
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 𝐹(𝐴) notation for a
function's value at 𝐴, i.e. "𝐹 of 𝐴,"
but without
context-dependent notational ambiguity. Alternate definitions are
dffv2 6271, dffv3 6187, fv2 6186,
and fv3 6206 (the latter two previously
required 𝐴 to be a set.) Restricted
equivalents that require 𝐹
to be a function are shown in funfv 6265 and funfv2 6266. For the familiar
definition of function value in terms of ordered pair membership, see
funopfvb 6239. (Contributed by NM, 1-Aug-1994.) Revised
to use
℩. Original version is now theorem dffv4 6188. (Revised by Scott
Fenton, 6-Oct-2017.) |
