| Description: Define the value of a
function, (𝐹‘𝐴), also known as function
application. For example, (cos‘0) = 1 (we
prove this in cos0 16344
after we define cosine in df-cos 16262). Typically, function 𝐹 is
defined using maps-to notation (see df-mpt 5190 and df-mpo 7445), but this is
not required. For example,
𝐹 =
{⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 31071).
Note that df-ov 7443 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 6937 and fvprc 6897).
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 7000, dffv3 6901, fv2 6900,
and fv3 6923 (the latter two previously
required 𝐴 to be a set.) Restricted
equivalents that require 𝐹
to be a function are shown in funfv 6992 and funfv2 6993. For the familiar
definition of function value in terms of ordered pair membership, see
funopfvb 6959. (Contributed by NM, 1-Aug-1994.) Revised
to use
℩. Original version is now Theorem dffv4 6902. (Revised by Scott
Fenton, 6-Oct-2017.) |
