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 $\phi': R \to \Img R$

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

Then an Epimorphism from Division Ring to Ring is either null or an isomorphism.

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