Cardinality of Set of Induced Equivalence Classes of Injection

Theorem
Let $f: S \to T$ be a mapping.

Let $\mathcal R_f \subseteq S \times S$ be the relation induced by $f$:
 * $\left({s_1, s_2}\right) \in \mathcal R_f \iff f \left({s_1}\right) = f \left({s_2}\right)$

Let $f$ be an injection.

Then there are $\left|{S}\right|$ different $\mathcal R_f$-classes.

Proof
From Cardinality of Image of Injection we have that $\left|{f \left({S}\right)}\right| = \left|{S}\right|$.

From the nature of an injection, for all $s \in S$, the $\mathcal R_f$-class of $s$ is a singleton.

Hence the result.