Image of Inverse Image

Theorem
Let $S, T$ be sets.

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

Let $X$ be a subset of $T$.

Then:
 * $f\left[{f^{-1}\left[{X}\right]}\right] \subseteq X$

where:
 * $f^{-1}\left[{X}\right]$ denotes the image of $X$ under the relation $f^{-1}$.

Proof
Let $x \in f \left[{f^{-1} \left[{X}\right]}\right]$.

By definition of image of set:
 * $\exists y \in S: y \in f^{-1} \left[{X}\right] \land x = f \left({y}\right)$

By definition of image of set under relation:
 * $f \left({y}\right) \in X$

Thus $x \in X$