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, $\forall x \in S: I_S \left({x}\right) = x$.

So $I_S \left({x}\right) = I_S \left({y}\right) \implies x = y$.

So, from the definition of an injection, $I_S$ is an injection.