Right Inverse Mapping is Injection

Theorem
A right inverse of a surjection is an injection.

Proof
Let $f: S \to T$ be a surjection.

Let $g: T \to S$ be a right inverse of $f$.

That is, let $f \circ g = I_T$, the identity mapping on $T$.

We have that $I_T$ is an injection.

By Injection if Composite is Injection, it follows that $g$ is an injection.

Also see

 * Left Inverse Mapping is Surjection