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:
 * $f \sqbrk {f^{-1} \sqbrk X} \subseteq X$

where:
 * $f^{-1} \sqbrk X$ denotes the image of $X$ under the relation $f^{-1}$.

Proof
Let $x \in f \sqbrk {f^{-1} \sqbrk X}$.

By definition of image of set:
 * $\exists y \in S: y \in f^{-1} \sqbrk X \land x = \map f y$

By definition of image of set under relation:
 * $\map f y \in X$

Thus $x \in X$