Originally Posted by

**Plato** $\displaystyle u \in \left( { \cap _\alpha f^{ - 1} (B_\alpha )} \right) \Leftrightarrow \left( {\forall \alpha \in A} \right)\left[ {u \in f^{ - 1} (B_\alpha )} \right]$

$\displaystyle \begin{gathered}

\Leftrightarrow \left( {\forall \alpha \in A} \right)\left[ {f(u) \in (B_\alpha )} \right] \hfill \\

\Leftrightarrow \left[ {f(u) \in \cap _\alpha (B_\alpha )} \right] \hfill \\

\Leftrightarrow u \in f^{ - 1} \left( { \cap _\alpha (B_\alpha )} \right) \hfill \\

\end{gathered} $

That is a start. Can you do the details?