# generator of sigma-field

I have an enlarged sigma-field $\sigma(\mathcal{F}\vee \sigma(X))$ (that is the smallest sigma-field containing the sigma-fields $\mathcal{F} \text{ and }\sigma(X)$?
Is this sigma-field generated by sets of the form $\{A \cap B: A \in\mathcal{F},B \in \sigma(X)\}$,
$\sigma(\{A \cap B: A \in\mathcal{F},B \in \sigma(X)\})=\sigma(\mathcal{F}\vee \sigma(X))$?