| Description: Define the value of a
function, (𝐹‘𝐴), also known as function
application. For example, (cos‘0) = 1 (we
prove this in cos0 14364
after we define cosine in df-cos 14284). Typically, function 𝐹 is
defined using maps-to notation (see df-mpt 4495 and df-mpt2 6368), but this
is not required. For example,
𝐹 =
{⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 26053).
Note that df-ov 6366 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 5952 and fvprc 5921).
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 6005, dffv3 5923, fv2 5922,
and fv3 5940 (the latter two previously
required 𝐴 to be a set.) Restricted
equivalents that require 𝐹
to be a function are shown in funfv 5999 and funfv2 6000. For the familiar
definition of function value in terms of ordered pair membership, see
funopfvb 5973. (Contributed by NM, 1-Aug-1994.) Revised
to use
℩. Original version is now theorem dffv4 5924. (Revised by Scott
Fenton, 6-Oct-2017.) |
