Generated Sigma-Algebra Preserves Subset

Theorem
Let $X$ be a set.

Let $\FF, \GG \subseteq \powerset X$ be collections of subsets of $X$.

Suppose that $\FF \subseteq \GG$.

Then $\map \sigma \FF \subseteq \map \sigma \GG$, where $\map \sigma \GG$ denotes the $\sigma$-algebra generated by $\GG$

Proof
By definition of $\map \sigma \GG$, $\GG \subseteq \map \sigma \GG$.

It follows that also $\FF \subseteq \map \sigma \GG$.

Hence, by definition of $\map \sigma \FF$, $\map \sigma \FF \subseteq \map \sigma \GG$.