| Description: Define the value of a
function, (𝐹‘𝐴), also known as function
application. For example, (cos‘0) = 1 (we
prove this in cos0 16112
after we define cosine in df-cos 16030). Typically, function 𝐹 is
defined using maps-to notation (see df-mpt 5156 and df-mpo 7364), but this is
not required. For example,
𝐹 =
{⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 30533).
Note that df-ov 7362 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 6862 and fvprc 6822).
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 6925, dffv3 6826, fv2 6825,
and fv3 6848 (the latter two previously
required 𝐴 to be a set.) Restricted
equivalents that require 𝐹
to be a function are shown in funfv 6917 and funfv2 6918. For the familiar
definition of function value in terms of ordered pair membership, see
funopfvb 6884. (Contributed by NM, 1-Aug-1994.) Revised
to use
℩. Original version is now Theorem dffv4 6827. (Revised by Scott
Fenton, 6-Oct-2017.) |
