Fixed Point of Idempotent Mapping

Theorem
Let $S$ be a set.

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

Let $f \left[{S}\right]$ be the image of $S$ under $f$.

Let $x \in S$.

Then $x$ is a fixed point of $f$ iff $x \in f \left[{S}\right]$.

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

Then:
 * $f \left({x}\right) = x$

and so by the image:
 * $x \in f \left[{S}\right]$

Sufficient Condition
Let $x \in f \left[{S}\right]$.

Then by the definition of image:


 * $\exists y \in S: f \left({y}\right) = x$

Then:

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