Mapping Induces Partition on Domain

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

Let $F$ be defined as:
 * $F = \left\{{f^{-1} \left({x}\right): x \in T}\right\}$

where $f^{-1} \left({x}\right)$ is the preimage of $x$.

Then $F$ is a partition of $S$.

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

Then from Induced Equivalence is an Equivalence Relation, $\mathcal R_f$ is an equivalence relation.

The result follows from Relation Partitions a Set iff Equivalence.