Element in Image of Preimage under Mapping

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

Then:
 * $\forall y \in T: \in f \left[{f^{-1} \left[{y}\right]}\right] = \left\{{y}\right\}$

Proof
A mapping is by definition a relation.

Therefore Image of Preimage of Mapping: Corollary applies:
 * $B \subseteq \operatorname{Im} \left({S}\right) \implies \left({f \circ f^{-1}}\right) \left[{B}\right] = B$

Thus:
 * $\left\{{y}\right\} \subseteq T \implies f^{-1} \left[{f \left[{\left\{{y}\right\}}\right]}\right] = \left\{{y}\right\}$

Hence the result.