Homomorph-containment is strongly join-closed

Statement with symbols
Suppose $$G$$ is a group, $$I$$ is an indexing set, and $$H_i, i \in I$$, is a collection of homomorph-containing subgroups of $$G$$. Then, the join of subgroups $$\langle H_i \rangle_{i \in I}$$ is also a homomorph-containing subgroup of $$G$$.

Related facts about join-closed

 * Isomorph-containment is strongly join-closed
 * Isomorph-freeness is strongly join-closed
 * Full invariance is strongly join-closed
 * Characteristicity is strongly join-closed
 * Normality is strongly join-closed

Proof
Given: A group $$G$$, an indexing set $$I$$, a collection $$H_i$$ of homomorph-containing subgroups of $$G$$, $$i \in I$$. $$H = \langle H_i \rangle_{i \in I}$$. A homomorphism $$\varphi: H \to G$$.

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

Proof: $$\varphi(H) = \varphi \langle H_i \rangle_{i \in I} = \langle \varphi(H_i) \rangle_{i \in I}$$. Since each $$H_i$$ is homomorph-containing in $$G$$, $$\varphi(H_i)$$ is contained in $$H_i$$, so the join of $$\varphi(H_i), i \in I$$, is contained in the join of the $$H_i$$s, which is $$H$$.