Preimage of Lower Set under Increasing Mapping is Lower

Theorem
Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be preordered sets.

Let $f: S \to T$ be an increasing mapping.

Let $X \subseteq T$ be a lower subset of $T$.

Then $f^{-1} \sqbrk X$ is lower

where $f^{-1} \sqbrk X$ denotes the preimage of $X$ under $f$.

Proof
Let $x \in f^{-1} \sqbrk X$, $y \in S$ such that $y \preceq x$.

Then: