Fixed Point of Idempotent Mapping

Theorem
Let $S$ be a set.

Let $f: S \to S$ be an idempotent mapping.

Let $x \in S$.

Then $x$ is a fixed point of $f$ iff $x \in f(S)$, where $f(S)$ is the image of $S$ under $f$.

Proof
Suppose that $x$ is a fixed point of $f$.

Then $f(x) = x$.

Thus $x \in f(S)$.

Suppose instead that $x \in f(S)$.

Then by the definition of image:
 * For some $y \in S$: $f(y) = x$

Since $f$ is idempotent:
 * $f(x) = f(f(y)) = f(y) = x$

Thus $x$ is a fixed point of $f$.