Quotient Set Determined by Relation Induced by Partition is That Partition

Theorem
Let $S$ be a set.

Let $\PP$ be a partition of $S$.

Let $\RR$ be the relation induced by $\PP$.

Then the quotient set $S / \RR$ of $S$ is $\PP$ itself.

Proof
Let $P \subseteq S$ such that $P \in \PP$.

Let $x \in P$.

Then:

Therefore:
 * $P = \eqclass x \RR$

and so:
 * $P \in S / \RR$

and so:
 * $\PP \subseteq S / \RR$

Now let $x \in S$.

As $\PP$ is a partition:
 * $\exists P \in \PP: x \in P$

Then by definition of $\RR$:
 * $\tuple {x, y} \in \RR \iff y \in \eqclass x \RR$

Therefore:
 * $\eqclass x \RR = P$

and so:
 * $\eqclass x \RR \in \PP$

That is:
 * $\SS / \RR \subseteq \PP$

It follows by definition of set equality that:
 * $\SS / \RR = P$

Hence the result.