Description: Define the value of a
function,      ,
also known as function
application. For example,     (we prove this in cos0 12743
after we define cosine in df-cos 12665). Typically, function is
defined using maps-to notation (see df-mpt 4260 and df-mpt2 6078), but this
is not required. For example,
         
(ex-fv 21743).
Note that df-ov 6076 will define two-argument functions using
ordered pairs
as  . 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 5747 and fvprc 5714).
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 notation for a
function's value at , i.e. " of ,"
but without
context-dependent notational ambiguity. Alternate definitions are
dffv2 5788, dffv3 5716, fv2 5715,
and fv3 5736 (the latter two previously
required to be
a set.) Restricted equivalents that require
to be a function are shown in funfv 5782 and funfv2 5783. For the familiar
definition of function value in terms of ordered pair membership, see
funopfvb 5762. (Contributed by NM, 1-Aug-1994.) Revised
to use
 .
Original version is now theorem dffv4 5717. (Revised by Scott
Fenton, 6-Oct-2017.) |
