Factoring Mapping into Quotient and Injection

Theorem
Any mapping $f: S \to T$ can be uniquely factored into a quotient mapping, followed by an injection.

Thus:
 * $f = h \circ q_{\mathcal R_f}$

where:


 * $q_{\mathcal R_f}: S \to S / \mathcal R_f : q_{\mathcal R_f} \left({s}\right) = \left[\!\left[{s}\right]\!\right]_{\mathcal R_f}$
 * $h: S / \mathcal R_f \to T : h \left({\left[\!\left[{s}\right]\!\right]_{\mathcal R_f}}\right) = f \left({s}\right)$

This can be illustrated using a commutative diagram as follows:


 * QuotientInjection.png

Proof
The mapping $q_{\mathcal R_f}: S \to S / \mathcal R_f$ follows from the definition of quotient mapping.

The mapping $h$ is justified by Existence of Renaming Mapping.

Also see

 * Factoring Mapping into Surjection and Inclusion


 * Quotient Theorem for Surjections
 * Quotient Theorem for Sets