| Description: Define the value of a
function, (𝐹‘𝐴), also known as function
application. For example, (cos‘0) = 1 (we
prove this in cos0 16051
after we define cosine in df-cos 15969). Typically, function 𝐹 is
defined using maps-to notation (see df-mpt 5171 and df-mpo 7346), but this is
not required. For example,
𝐹 =
{⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 30413).
Note that df-ov 7344 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 6849 and fvprc 6809).
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 6912, dffv3 6813, fv2 6812,
and fv3 6835 (the latter two previously
required 𝐴 to be a set.) Restricted
equivalents that require 𝐹
to be a function are shown in funfv 6904 and funfv2 6905. For the familiar
definition of function value in terms of ordered pair membership, see
funopfvb 6871. (Contributed by NM, 1-Aug-1994.) Revised
to use
℩. Original version is now Theorem dffv4 6814. (Revised by Scott
Fenton, 6-Oct-2017.) |
