Quotient Theorem for Surjections

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Theorem

For any surjection $f: S \to T$, there is one and only one bijection $r: S / \mathcal R_f \to T$ such that:

$r \circ q_{\mathcal R_f} = f$

where:

$\mathcal R_f$ is the equivalence induced by $f$;
$r: S / \mathcal R_f \to T$ is the renaming mapping;
$q_{\mathcal R_f}: S \to S / \mathcal R_f$ is the quotient mapping induced by $\mathcal R_f$.

Also known as the factor theorem for surjections.

From the definition of Induced Equivalence, the mapping $f: S \to T$ induces an equivalence $\mathcal R_f$ on its domain.

As $f: S \to T$ is a surjection, $T = \operatorname{Im} \left({f}\right)$ from Surjection iff Image equals Codomain.

From Renaming Mapping is a Bijection, the renaming mapping $r: S / \mathcal R_f \to T$ is a bijection, where $S / \mathcal R_f$ is the quotient set of $S$ by $\mathcal R_f$.

Clearly:

$r \circ q_{\mathcal R_f} = f$.
$r$ is the only mapping $r: S / \mathcal R_f \to T$ that satisfies this equality.

$\blacksquare$