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.