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 $\sigma$-algebra generated by $\GG$:
 * $\GG \subseteq \map \sigma \GG$

It follows that also:
 * $\FF \subseteq \map \sigma \GG$

Hence, by definition of $\sigma$-algebra generated by $\FF$:
 * $\map \sigma \FF \subseteq \map \sigma \GG$