| Description: Define the value of a
function, (𝐹‘𝐴), also known as function
application. For example, (cos‘0) = 1 (we
prove this in cos0 15282
after we define cosine in df-cos 15203). Typically, function 𝐹 is
defined using maps-to notation (see df-mpt 4966 and df-mpt2 6927), but this
is not required. For example,
𝐹 =
{⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 27889).
Note that df-ov 6925 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 6476 and fvprc 6439).
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 6531, dffv3 6442, fv2 6441,
and fv3 6464 (the latter two previously
required 𝐴 to be a set.) Restricted
equivalents that require 𝐹
to be a function are shown in funfv 6525 and funfv2 6526. For the familiar
definition of function value in terms of ordered pair membership, see
funopfvb 6498. (Contributed by NM, 1-Aug-1994.) Revised
to use
℩. Original version is now theorem dffv4 6443. (Revised by Scott
Fenton, 6-Oct-2017.) |
