Image of Directed Subset under Increasing Mapping is Directed

Theorem
Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets.

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

Let $D$ be a directed subset of $S$.

Then
 * $f^\to\left({D}\right)$ is also a directed subset of $T$

where
 * $f^\to\left({D}\right)$ denotes the image of $D$ under $f$.

Proof
Let $x, y \in f^\to\left({D}\right)$.

By definition of image of set:
 * $\exists a \in D: x = f\left({a}\right)$

and
 * $\exists b \in D: y = f\left({b}\right)$

By definition of directed subset:
 * $\exists c \in D: a \preceq c \land b \preceq c$

By definition of image of set:
 * $f\left({c}\right) \in f^\to\left({D}\right)$

By definition of increasing mapping:
 * $x \precsim f\left({c}\right)$ and $y \precsim f\left({c}\right)$

Thus by definition
 * $f^\to\left({D}\right)$ is a directed subset of $T$.