How do I prove that if something is true for $\sigma$-algebra generators then it's true for the algebra?

How do I prove that if something is true for $\sigma$-algebra generators
then it's true for the algebra?

Let $\mathcal{F}_1 \otimes \mathcal{F}_2$ be the product sigma algebra.
It's generated by the measurable rectangles $A\times B$, where $A \in
\mathcal{F}_1$ and $B \in \mathcal {F_2}$. For an example of my question,
let
$$ \mathcal{D} = \{ E \in \mathcal{F}_1 \otimes \mathcal{F}_2 : \mu(E) =
(\mu_1\otimes\mu_2)(E)\} $$ Then the book asks me to prove that it's a
Dynkin system. Now I've worked through examples before that were able to
prove such things without the "works on generators then works on the
generated set" method, but it seems like I'm constantly running into these
situations, so I would like to learn that method of proof. So for now how
would I prove that if Dynkin system definition holds on measurable
rectangles in $\mathcal{D}$ then it holds for all of $\mathcal{D}$.
Thanks.
I'm given that the set-defining property above holds for measurable
rectangles in $\mathcal{F}_1 \times \mathcal{F}_2$.