How to make functional rules not depend on an argument?

Question

Consider the following:

rule = f[x] -> Sin[x]

if I now do, say:

f[y] /. rule

this does not work, because the rule wants the argument to specifically be x. Of course, I could do:

f[x] /. rule /. x -> y

but I really would be able to "clean" the rule of the input. As an example, If we use NDSolve to solve an ODE, we obtain a rule of the form:

{f -> InterpolatingFunction["stuff"]}

this is nice insofar that I can apply this rule to f, where the argument can be anything I wish. So, the question is, If I ALREADY have a rule as above, is there a nice way to "remove" the "[x]" part of all the functions, in the sense described above?

Answer 1

While John Doty's answer works pretty well, if you want to obtain a pure function as rhs of your rule like in your InterpolatingFunction example, you can do something like

rule = f[x] -> Sin[x];
newrule = f -> Block[
 {x},
 Function @@ List[{x}, f[x] /. rule]
]
(*f -> Function[{x}, Sin[x]]*)

Answer 2

It's just like defining a function. Make x the name of a pattern, not a literal symbol:

rule = f[x_] -> Sin[x]
f[y] /. rule
(* Sin[y] *)