Cardinality of Set of Induced Equivalence Classes of Surjection

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 a surjection.

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

Proof
From the definition of a surjection:
 * $\forall t \in T: \exists s \in S: f \left({s}\right) = t$

Thus there are as many $\mathcal R_f$-classes of $f$ as there are elements of $T$.

Hence the result.