Left Operation is Closed for All Subsets

Theorem
Let $S$ be a set.

Let $\leftarrow$ be the left operation on $S$.

That is:
 * $\forall x, y \in S: x \leftarrow y = x$

Let $\powerset S$ be the power set of $S$.

Then for all $T \in \powerset S$, $\leftarrow$ is closed on $T$.

That is, for all $T \in \powerset S$:
 * $\struct {T, \leftarrow}$ is a subsemigroup of $\struct {S, \leftarrow}$.

Proof
From Element under Left Operation is Right Identity we have that $\struct {S, \leftarrow}$ is a semigroup, whatever the nature of $S$.

Let $T \in \powerset S$.

Then:
 * From Element under Left Operation is Right Identity, $\struct {T, \leftarrow}$ is a semigroup, and therefore a subsemigroup of $\struct {S, \leftarrow}$.

This applies whatever $S$ is and whatever the subset $T$ is.

Also see

 * Right Operation is Closed for All Subsets