Operator Generated by Image of Closure Operator is Closure Operator

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.

Let $c:S \to S$ be a closure operator on $L$.

Then $\operatorname{operator}\left({\left({c\left[{S}\right], \precsim}\right)}\right) = c$

where
 * $\mathord\precsim = \mathord\preceq \cap \left({c\left[{S}\right] \times c\left[{S}\right]}\right)$
 * $\operatorname{operator}\left({\left({c\left[{S}\right], \precsim}\right)}\right) =$ denotes the operator generated by $\left({c\left[{S}\right], \precsim}\right)$

Proof
Let $x \in S$.

By definition of closure operator/inflationary:
 * $x \preceq c\left({x}\right)$

By definition of upper closure of element:
 * $c\left({x}\right) \in x^\succeq$

By definition of image of mapping:
 * $c\left({x}\right) \in c\left[{S}\right]$

By definition of intersection:
 * $c\left({x}\right) \in x^\succeq \cap c\left[{S}\right]$

By definitions of infimum and lower bound:
 * $\inf_L\left({x^\succeq \cap c\left[{S}\right]}\right) \preceq c\left({x}\right)$

We will prove that
 * $c\left({x}\right)$ is lower bound for $x^\succeq \cap c\left[{S}\right]$

Let $y \in x^\succeq \cap c\left[{S}\right]$

By definition of intersection:
 * $y \in x^\succeq$ and $y \in c\left[{S}\right]$

By definition of image of mapping:
 * $\exists z \in S: y = c\left({z}\right)$

By definition of closure operator/idempotent:
 * $y = c\left({y}\right)$

By definition of upper closure of element:
 * $x \preceq y$

Thus by definition of closure operator/increasing:
 * $c\left({x}\right) \preceq y$

By definition of infimum:
 * $c\left({x}\right) \preceq \inf_L\left({x^\succeq \cap c\left[{S}\right]}\right)$

By definition of antisymmetry:
 * $c\left({x}\right) = \inf_L\left({x^\succeq \cap c\left[{S}\right]}\right)$

Thus by definition of operator generated by $\left({c\left[{S}\right], \precsim}\right)$:
 * $\operatorname{operator}\left({\left({c\left[{S}\right], \precsim}\right)}\right)\left({x}\right) = c\left({x}\right)$

Hence by Equality of Mappings:
 * $\operatorname{operator}\left({\left({c\left[{S}\right], \precsim}\right)}\right) = c$