Let $S$ be a set. We say $S$ is $D^\ast$[i]-finite[/i] if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$, there exists a $y \in Y$ such that $f(y) \notin Y$.  The function $f$ is called a [i]witness[/i] of $S$.  How many witnesses does $\{0,1,\cdots,5\}$ have?

[i]Proposed by Evan Chen[/i]