Quotient Mapping is Injection iff Equality

Theorem
Let $\RR$ be an equivalence relation on $S$.

Then the quotient mapping $q_\RR: S \to S / \RR$ is an injection $\RR$ is the equality relation.

Proof
Let $\eqclass x \RR, \eqclass y \RR \in S / \RR$

Sufficient Condition
Let $q_\RR: S \to S / \RR$ be an injection.

Then:

That is:
 * $\RR$ is the equality relation.

Necessary Condition
Let $q_\RR: S \to S / \RR$ be a mapping which is specifically not an injection.

Then:

That is:
 * $a \ne b$

but:
 * $a \mathrel \RR b$

and so $\RR$ is not the equality relation.

From Rule of Transposition it follows that:
 * if $\RR$ is the equality relation then $q_\RR$ is an injection.