| Description: Define the value of a
function, (𝐹‘𝐴), also known as function
application. For example, (cos‘0) = 1 (we
prove this in cos0 16184
after we define cosine in df-cos 16102). Typically, function 𝐹 is
defined using maps-to notation (see df-mpt 5184 and df-mpo 7403), but this is
not required. For example,
𝐹 =
{⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 30647).
Note that df-ov 7401 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 6901 and fvprc 6861).
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 6964, dffv3 6865, fv2 6864,
and fv3 6887 (the latter two previously
required 𝐴 to be a set.) Restricted
equivalents that require 𝐹
to be a function are shown in funfv 6956 and funfv2 6957. For the familiar
definition of function value in terms of ordered pair membership, see
funopfvb 6923. (Contributed by NM, 1-Aug-1994.) Revised
to use
℩. Original version is now Theorem dffv4 6866. (Revised by Scott
Fenton, 6-Oct-2017.) |
