Infimum of Image of Upper Closure of Element under Increasing Mapping

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

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

Let $x \in S$.

Then $\map \inf {f \sqbrk {x^\succeq} } = \map f x$

Proof
By Infimum of Upper Closure of Element:
 * $\inf x^\succeq = x$

By definition of infimum:
 * $x$ is lower bound for $x^\succeq$

Thus by Increasing Mapping Preserves Lower Bounds:
 * $\map f x$ is lower bound for $f \sqbrk {x^\succeq}$

By definition of reflexivity:
 * $x \preceq x$

By definition of upper closure of element:
 * $x \in x^\succeq$

By definition of image of set:
 * $\map f x \in f \sqbrk {x^\succeq}$

Thus by definition:
 * $\forall y \in T: y$ is lower bound for $f \sqbrk {x^\succeq} \implies y \precsim \map f x$

Thus by definition of infimum:
 * $\map \inf {f \sqbrk {x^\succeq} } = \map f x$