Galois Connection is Expressed by Minimum

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

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

Then $\left({g, d}\right)$ is Galois connection
 * $g$ is increasing mapping and
 * $\forall t \in T: d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$

where
 * $\min$ denotes the minimum
 * $g^{-1}\left[{t^\succsim}\right]$ 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 imsge 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 ainfimum

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

Thus