Saturation Under Equivalence Relation in Terms of Graph

Theorem
Let $\RR \subset S \times S$ be an equivalence relation on a set $S$.

Let $\pr_1, \pr_2 : S \times S \to S$ denote the projections.

Let $T\subset S$ be a subset.

Let $\overline T$ denote its saturation.

Then the following hold:
 * $\overline T = \map {\pr_1} {\RR \cap \map {\pr_2^{-1} } T}$
 * $\overline T = \map {\pr_2} {\RR \cap \map {\pr_1^{-1} } T}$

Proof
Let $s \in S$.

We have:

A similar reasoning proves the second identity.