Galois Connection is Expressed by Minimum

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

Let $g: S \to T$, $d: T \to S$ be mappings.

Then $\tuple {g, d}$ is a Galois connection :
 * $g$ is an increasing mapping and
 * $\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {t^\succsim} }$

where
 * $\min$ denotes the minimum
 * $g^{-1} \sqbrk {t^\succsim}$ denotes the image of $t^\succsim$ under relation $g^{-1}$
 * $t^\succsim$ denotes the upper closure of $t$

Sufficient Condition
Let $\left({g, d}\right)$ be a Galois connection.

Thus by definition of Galois connection:
 * $g$ is an increasing mapping

Let $t \in T$.

By definition of reflexivity:
 * $d\left({t}\right) \preceq d\left({t}\right)$

By definition of Galois connection:
 * $t \precsim g\left({d\left({t}\right)}\right)$

By definition of upper closure:
 * $g\left({d\left({t}\right)}\right) \in t^\succsim$

By definition of image of set under relation:
 * $d\left({t}\right) \in g^{-1}\left[{t^\succsim}\right]$

By definition of lower bound:
 * $\forall s \in S: s$ is lower bound for $g^{-1}\left[{t^\succsim}\right] \implies s \preceq d\left({t}\right)$

We will prove that
 * $d\left({t}\right)$ is lower bound for $g^{-1}\left[{t^\succsim}\right]$

Let $s \in g^{-1}\left[{t^\succsim}\right]$.

By definition of image of set:
 * $g\left({s}\right) \in t^\succsim$

By definition of upper closure of element:
 * $t \precsim g\left({s}\right)$

Thus by definition of Galois connection:
 * $d\left({t}\right) \preceq s$

By definition of infimum:
 * $g^{-1}\left[{t^\succsim}\right]$ admits an infimum

and
 * $\inf\left({g^{-1}\left[{t^\succsim}\right]}\right) = d\left({t}\right)$

Thus:
 * $d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$

Necessary Condition
Let $g: S \to T$ be an increasing mapping.

Let $\forall t \in T: d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$

Thus:
 * $g$ is increasing mapping.

We will prove that
 * $d$ is increasing mapping.

Let $x, y \in T$ such that
 * $x \precsim y$

By Upper Closure is Decreasing:
 * $y^\succsim \subseteq x^\succsim$

By Image of Subset under Relation is Subset of Image/Corollary 3:
 * $g^{-1}\left[{y^\succsim}\right] \subseteq g^{-1}\left[{x^\succsim}\right]$

By assumption
 * $d\left({x}\right) = \min\left({g^{-1}\left[{x^\succsim}\right]}\right) = \inf\left({g^{-1}\left[{x^\succsim}\right]}\right)$

and
 * $d\left({y}\right) = \min\left({g^{-1}\left[{y^\succsim}\right]}\right) = \inf\left({g^{-1}\left[{y^\succsim}\right]}\right)$

By Infimum of Subset:
 * $d\left({x}\right) \preceq d\left({y}\right)$

Thus by definition:
 * $d$ is an increasing mapping.

We will prove that
 * $\forall s \in S, t \in T: t \precsim g\left({s}\right) \iff d\left({t}\right) \preceq s$

Let $s \in S, t \in T$.

First implication:

Let $t \precsim g\left({s}\right)$.

By definition of upper closure of element:
 * $g\left({s}\right) \in t^\succsim$

By definition of image of set:
 * $s \in g^{-1}\left[{t^\succsim}\right]$

By assumption
 * $d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right) = \inf\left({g^{-1}\left[{t^\succsim}\right]}\right)$

By definition of infimum:
 * $d\left({t}\right)$ is lower bound for $g^{-1}\left[{t^\succsim}\right]$

Thus by definition of lower bound:
 * $d\left({t}\right) \preceq s$

Second implication:

Let $d\left({t}\right) \preceq s$

By assumption
 * $d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$

By smallest element of set:
 * $d\left({t}\right) \in g^{-1}\left[{t^\succsim}\right]$

By definition of image of set:
 * $g\left({d\left({t}\right)}\right) \in t^\succsim$

By definition of upper closure of element:
 * $t \precsim g\left({d\left({t}\right)}\right)$

By definition of increasing mapping:
 * $g\left({d\left({t}\right)}\right) \precsim g\left({s}\right)$

Thus by definition of transitivity:
 * $t \precsim g\left({s}\right)$

Thus by definition:
 * $\left({g, d}\right)$ is Galois connection.