Description: Define the value of a
function, (𝐹‘𝐴), also known as function
application. For example, (cos‘0) = 1 (we
prove this in cos0 14719
after we define cosine in df-cos 14640). Typically, function 𝐹 is
defined using maps-to notation (see df-mpt 4645 and df-mpt2 6554), but this
is not required. For example,
𝐹 =
{〈2, 6〉, 〈3, 9〉} → (𝐹‘3) = 9 (ex-fv 26692).
Note that df-ov 6552 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 6128 and fvprc 6097).
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 6181, dffv3 6099, fv2 6098,
and fv3 6116 (the latter two previously
required 𝐴 to be a set.) Restricted
equivalents that require 𝐹
to be a function are shown in funfv 6175 and funfv2 6176. For the familiar
definition of function value in terms of ordered pair membership, see
funopfvb 6149. (Contributed by NM, 1-Aug-1994.) Revised
to use
℩. Original version is now theorem dffv4 6100. (Revised by Scott
Fenton, 6-Oct-2017.) |