Ordered Set of All Mappings is Ordered Set

Theorem

Let $L = \left({S, \preceq}\right)$ be an ordered set.

Let $X$ be a set.

Then $L^X$ is also an ordered set.

Proof

By definition of ordered set of all mappings:

$L^X = \left({S^X, \precsim}\right)$

where

$\forall f, g \in S^X: f \precsim g \iff f \preceq g$
$\preceq$ denotes the ordering on mappings,
$S^X$ denotes the set of all mappings from $X$ into $S$.

Reflexivity

Let $f \in S^X$.

By definition of reflexivity:

$\forall x \in X: f\left({x}\right) \preceq f\left({x}\right)$

By definition of ordering on mappings:

$f \preceq f$

Thus by definition of $\precsim$:

$f \precsim f$

$\Box$

Transitivity

Let $f, g, h \in S^X$ such that

$f \precsim g$ and $g \precsim h$

By definition of $\precsim$:

$f \preceq g$ and $g \preceq h$

By definition of ordering on mappings:

$\forall x \in X: f\left({x}\right) \preceq g\left({x}\right)$

and

$\forall x \in X: g\left({x}\right) \preceq h\left({x}\right)$

By definition of transitivity:

$\forall x \in X: f\left({x}\right) \preceq h\left({x}\right)$

By definition of ordering on mappings:

$f \preceq h$

Thus by definition of $\precsim$:

$f \precsim h$

$\Box$

Antisymmetry

Let $f, g \in S^X$ such that

$f \precsim g$ and $g \precsim f$

By definition of $\precsim$:

$f \preceq g$ and $g \preceq f$

By definition of ordering on mappings:

$\forall x \in X: f\left({x}\right) \preceq g\left({x}\right)$

and

$\forall x \in X: g\left({x}\right) \preceq f\left({x}\right)$

By definition of antisymmetry:

$\forall x \in X: f\left({x}\right) = g\left({x}\right)$

Thus by Equality of Mappings:

$f = g$

$\blacksquare$