Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism/Proof 2

Proof
From Surjection by Restriction of Codomain, we can restrict the codomain of $\phi$ and consider the mapping $\phi': R \to \operatorname {Im} \left({R}\right)$

As $\phi'$ is now a surjective homomorphism, it is by definition an epimorphism.

Then we invoke the result that an epimorphism from a division ring to a ring is either null or an isomorphism.

As an isomorphism is by definition injective, the result follows.