Suprema Preserving Mapping on Ideals is Increasing

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

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

For every ideal $I$ in $\struct {S, \preceq}$, let $f$ preserve the supremum on $I$.

Then $f$ is increasing.

Proof
Let $x, y \in S$ such that:
 * $x \preceq y$

By Supremum of Singleton:
 * $\set x$ and $\set y$ admit suprema in $\struct {S, \preceq}$

By Supremum of Lower Closure of Set:
 * $\set x^\preceq$ and $\set y^\preceq$ admit suprema in $\struct {S, \preceq}$

where $\set x^\preceq$ denotes the lower closure of $\set x$

By Lower Closure of Singleton:
 * $x^\preceq$ and $y^\preceq$ admit suprema in $\struct {S, \preceq}$

By Lower Closure of Element is Ideal:
 * $x^\preceq$ and $y^\preceq$ are ideals in $\struct {S, \preceq}$

By assumption and definition of mapping preserves the supremum on subset:
 * $\map {f^\to} {x^\preceq}$ and $\map {f^\to} {y^\preceq}$ admit suprema in $\struct {T, \precsim}$

and:
 * $\map \sup {\map {f^\to} {x^\preceq} } = \map f {\map \sup {x^\preceq} }$

and:
 * $\map \sup {\map {f^\to} {y^\preceq} } = \map f {\map \sup {y^\preceq} }$

By Supremum of Lower Closure of Element:
 * $\map \sup {x^\preceq} = x$ and $\map \sup {y^\preceq} = y$

By Lower Closure is Increasing:
 * $x^\preceq \subseteq y^\preceq$

By Image of Subset under Mapping is Subset of Image:
 * $\map {f^\to} {x^\preceq} \subseteq \map {f^\to} {y^\preceq}$

Thus by Supremum of Subset:
 * $\map f x \precsim \map f y$

Thus by definition:
 * $f$ is increasing.