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.

$\blacksquare$

Sources

• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.14$: Composition of Functions: Theorem $14.7$