Ring Homomorphism Preserves Subrings/Proof 2

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

If $S$ is a subring of $R_1$, then $\phi \left({S}\right)$ is a subring of $R_2$.

Proof
From Morphism Property Preserves Closure, $\phi \left({R_1}\right)$ is a closed algebraic structure.

From Epimorphism Preserves Rings, $\phi \left({S}\right)$ is a ring.

Hence the result, from the definition of subring.