Saturation Under Equivalence Relation in Terms of Graph

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

Let $\operatorname{pr}_1,\operatorname{pr}_2 : S \times S \to S$ denote the projections.

Let $T\subset S$ be a subset.

Let $\overline T$ denote its saturation.

Then the following hold:
 * $\overline T = \operatorname{pr}_1(\mathcal R \cap\operatorname{pr}_2^{-1}(T))$
 * $\overline T = \operatorname{pr}_2(\mathcal R \cap\operatorname{pr}_1^{-1}(T))$

Proof
Let $s\in S$.

We have:

A similar reasoning proves the second identity.