Identity Mapping is Idempotent

Theorem
Let $S$ be a set.

Let $I_S: S \to S$ be the identity mapping on $S$.

Then $I_S$ is idempotent:
 * $I_S \circ I_S = I_S$

Proof
From Identity Mapping is Left Identity:
 * $I_S \circ f = f$

for all mappings $f: S \to S$.

From Identity Mapping is Right Identity:
 * $f \circ I_S = f$

for all mappings $f: S \to S$.

Substituting $I_S$ for $f$ in either one and the result follows.