Identity Mapping is Injection

Theorem
On any set $S$, the identity mapping $I_S: S \to S$ is an injection.

Proof
From the definition of the identity mapping:
 * $\forall x \in S: \map {I_S} x = x$

So:
 * $\map {I_S} x = \map {I_S} y \implies x = y$

So by definition $I_S$ is an injection.

Also see

 * Identity Mapping is Surjection