Homomorph-containment satisfies intermediate subgroup condition

Statement

Verbal statement

A homomorph-containing subgroup of the whole group is also a homomorph-containing subgroup of any intermediate subgroup.

Statement with symbols

Suppose $H \le K \le G$ are groups, and $H$ is a homomorph-containing subgroup of $G$. Then, $H$ is also a homomorph-containing subgroup of $K$.

Proof

Given: Groups $H \le K \le G$ such that $H$ is homomorph-containing in $G$. A homomorphism $\varphi:H \to K$.

To prove: $\varphi(H)$ is contained in $H$.

Proof: Since $K \le G$, we can compose $\varphi$ with the inclusion of $K$ in $G$ to get a homomorphism $\varphi':H \to G$. Since $H$ is homomorph-containing in $G$, $\varphi'(H) \le H$, so $\varphi(H) \le H$.