Quotient Mapping is Injection iff Equality

Theorem
Let $\mathcal R$ be an equivalence relation on $S$.

Then the quotient mapping $q_{\mathcal R}: S \to S / \mathcal R$ is an injection $\mathcal R$ is the equality relation.

Proof
Let $\eqclass x {\mathcal R}, \eqclass y {\mathcal R} \in S / \mathcal R$

Sufficient Condition
Let $q_{\mathcal R}: S \to S / \mathcal R$ be an injection.

Then:

That is:
 * $\mathcal R$ is the equality relation.

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

Then:

That is:
 * $a \ne b$

but:
 * $a \mathrel {\mathcal R} b$

and so $\mathcal R$ is not the equality relation.

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