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:


 * $g: S \to \Img f: g \paren x = f \paren x$


 * $i_T: \Img f \to T: i_T \paren x = x$

This can be illustrated using a commutative diagram as follows:


 * $\begin{xy}\xymatrix@L+2mu@+1em {

S \ar@{-->}[r]^*{g} \ar[rd]_*{f = i_T \circ g} & \Img f \ar@{-->}[d]^*{i_T} \\ & T }\end{xy}$

Proof
From Surjection by Restriction of Codomain, any $f: S \to \Img f$ is a surjection.

The mapping $g: S \to \Img f$ where $\map g x = \map f x$ is therefore also clearly a surjection.

The mapping $g: S \to \Img f: \map g x = \map f x$ is clearly unique, by Equality of Mappings.

From Inclusion Mapping is Injection, $i_T: \Img f \to T$ is an injection.

Likewise, the mapping $i_T: \Img f \to T : \map {i_T} x = x$ is also unique, by its own definition.

Also see

 * Factoring Mapping into Quotient and Injection


 * Quotient Theorem for Surjections
 * Quotient Theorem for Sets