Ring Homomorphism Preserves Subrings

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

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

Then $\phi \sqbrk S$ is a subring of $R_2$.