Intermediately operator preserves join-closedness

Statement
Suppose $$p$$ is a subgroup property and $$I$$ is the fact about::intermediately operator. Then:


 * If $$p$$ is a fact about::join-closed subgroup property (respectively, fact about::strongly join-closed subgroup property), so is $$I(p)$$.
 * If $$p$$ is a fact about::finite-join-closed subgroup property (respectively, fact about::strongly finite-join-closed subgroup property), so is $$I(p)$$.

Applications

 * Intermediate characteristicity is strongly join-closed
 * Intermediate full invariance is strongly join-closed

Other similar facts

 * Intermediately operator preserves quotient-transitivity