Factoring Mapping into Surjection and Inclusion

Theorem
Every mapping $f:S \to T$ can be uniquely factored into a surjection $g$ followed by the inclusion mapping $i_T$.

That is, $f = i_T \circ g$ where:


 * $g: S \to \operatorname{Im} \left({f}\right) : g \left({x}\right) = f \left({x}\right)$
 * $i_T: \operatorname{Im} \left({f}\right) \to T : i_T \left({x}\right) = x$

This can be illustrated using a commutative diagram as follows:


 * SurjectionInclusion.png

Proof

 * From Surjection by Restriction of Codomain, any $f: S \to \operatorname{Im} \left({f}\right)$ is a surjection.

The mapping $g: S \to \operatorname{Im} \left({f}\right)$ where $g \left({x}\right) = f \left({x}\right)$ is therefore also clearly a surjection.

The mapping $g: S \to \operatorname{Im} \left({f}\right) : g \left({x}\right) = f \left({x}\right)$ is clearly unique, by Equality of Mappings.


 * From Inclusion Mapping is an Injection, $i_T: \operatorname{Im} \left({f}\right) \to T$ is an injection.

Likewise, the mapping $i_T: \operatorname{Im} \left({f}\right) \to T : i_T \left({x}\right) = x$ is also unique, by its own definition.

Also see

 * Factoring Mapping into Quotient and Injection


 * Quotient Theorem for Surjections
 * Quotient Theorem for Sets