Generated Sigma-Algebra Contains Generated Sigma-Algebra of Subset

Theorem
Let $\map \sigma \FF$ be the $\sigma$-algebra generated by $\EE$.

Let $\map \sigma \FF$ contain a set of sets $\EE$.

Let $\map \sigma \EE$ be the $\sigma$-algebra generated by $\EE$.

Then $\map \sigma \EE \subseteq \map \sigma \FF$.

Proof
Apply Sigma-Algebra Contains Generated Sigma-Algebra of Subset to $\map \sigma \FF$.