Full invariance is strongly join-closed

Statement
Suppose $$G$$ is a group, and $$H_i, i \in I$$ are fully invariant subgroups of $$G$$. Then, the join of subgroups $$\langle H_i \rangle$$ is also a fully invariant subgroup.

Related facts
A generalization is:

endo-invariance implies strongly join-closed

Other instances of the generalization are:


 * Normality is strongly join-closed
 * Characteristicity is strongly join-closed
 * Strict characteristicity is strongly join-closed

Facts used

 * 1) uses::Endo-invariance implies strongly join-closed

Proof
The proof follows directly from fact (1), since full invariance is an endo-invariance property.