| Description: Define the value of a
function, (𝐹‘𝐴), also known as function
application. For example, (cos‘0) = 1 (we
prove this in cos0 15086
after we define cosine in df-cos 15007). Typically, function 𝐹 is
defined using maps-to notation (see df-mpt 4864 and df-mpt2 6798), but this
is not required. For example,
𝐹 =
{⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 27642).
Note that df-ov 6796 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 6359 and fvprc 6326).
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 6413, dffv3 6328, fv2 6327,
and fv3 6347 (the latter two previously
required 𝐴 to be a set.) Restricted
equivalents that require 𝐹
to be a function are shown in funfv 6407 and funfv2 6408. For the familiar
definition of function value in terms of ordered pair membership, see
funopfvb 6380. (Contributed by NM, 1-Aug-1994.) Revised
to use
℩. Original version is now theorem dffv4 6329. (Revised by Scott
Fenton, 6-Oct-2017.) |
