Ring Epimorphism Preserves Subrings

Theorem
Let $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$ be a ring epimorphism.

Let $S$ be a subring of $R_1$.

Then $\phi \left({S}\right)$ is a subring of $R_2$.

Proof
A direct application of Ring Homomorphism Preserves Subrings.