Fixed Point of Idempotent Mapping

Theorem
Let $S$ be a set.

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

Let $\Img f$ be the image of $f$.

Let $x \in S$.

Then $x$ is a fixed point of $f$ $x \in \Img f$.

Necessary Condition
Let $x$ be a fixed point of $f$.

Then:
 * $\map f x = x$

and so by definition of image of mapping:
 * $x \in \Img f$

Sufficient Condition
Let $x \in \Img f$.

Then by the definition of image:


 * $\exists y \in S: \map f y = x$

Then:

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