Operator Generated by Image of Closure Operator is Closure Operator

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.

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

Then $\map {\operatorname {operator} } {\struct {c \sqbrk S, \precsim} } = c$

where
 * $\mathord \precsim = \mathord \preceq \cap \paren {c \sqbrk S \times c \sqbrk S}$
 * $\map {\operatorname {operator} } {\struct {c \sqbrk S, \precsim} }$ denotes the operator generated by $\struct {c \sqbrk S, \precsim}$

Proof
Let $x \in S$.

By definition of closure operator/inflationary:
 * $x \preceq \map c x$

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

By definition of image of mapping:
 * $\map c x \in c \sqbrk S$

By definition of intersection:
 * $\map c x \in x^\succeq \cap c \sqbrk S$

By definitions of infimum and lower bound:
 * $\map {\inf_L} {x^\succeq \cap c \sqbrk S} \preceq \map c x$

We will prove that
 * $\map c x$ is lower bound for $x^\succeq \cap c \sqbrk S$

Let $y \in x^\succeq \cap c \sqbrk S$

By definition of intersection:
 * $y \in x^\succeq$ and $y \in c \sqbrk S$

By definition of image of mapping:
 * $\exists z \in S: y = \map c z$

By definition of closure operator/idempotent:
 * $y = \map c y$

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

Thus by definition of closure operator/increasing:
 * $\map c x \preceq y$

By definition of infimum:
 * $\map c x \preceq \map {\inf_L} {x^\succeq \cap c \sqbrk S}$

By definition of antisymmetry:
 * $\map c x = \map {\inf_L} {x^\succeq \cap c \sqbrk S}$

Thus by definition of operator generated by $\struct {c \sqbrk S, \precsim}$:
 * $\map {\map {\operatorname {operator} } {\struct {c \sqbrk S, \precsim} } } x = \map c x$

Hence by Equality of Mappings:
 * $\map {\operatorname {operator} } {\struct {c \sqbrk S, \precsim} } = c$