Subset of Domain is Subset of Preimage of Image

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

Then:
 * $A \subseteq S \implies A \subseteq \left({f^{-1} \circ f}\right) \left({A}\right)$

Proof
As a mapping is by definition also a relation, we apply Preimage of Image directly:
 * $A \subseteq S \implies A \subseteq \left({\mathcal R^{-1} \circ \mathcal R}\right) \left({A}\right)$

where $\mathcal R$ is a relation.

Hence:
 * $A \subseteq S \implies A \subseteq \left({f^{-1} \circ f}\right) \left({A}\right)$