Element in Preimage of Image under Mapping

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

Then:
 * $\forall x \in S: x \in f^{-1} \left[{f \left({x}\right)}\right]$

Proof
A mapping is by definition a relation.

Therefore Preimage of Image applies:
 * $A \subseteq S \implies A \subseteq f^{-1} \left[{f \left[{A}\right]}\right]$

Thus:
 * $\left\{{x}\right\} \subseteq S \implies \left\{{x}\right\} \subseteq f^{-1} \left[{f \left[{A}\right]}\right]$

Hence the result.