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Math Help - Inverse of intersections

  1. #1
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    Inverse of intersections

    Prove that  f^{-1} ( \bigcap _ { \alpha \in A } ) = \bigcap _{ \alpha \in A } f^ {-1} (E_ \alpha )

    My proof.

    Pick  x \in f^{-1} ( \bigcap _ { \alpha \in A } E_ \alpha)

    implies that  \exists y \in \bigcap _{ \alpha \in A } E_ \alpha such that  f(y)=x

    Note that  y \in E_ \alpha \ \ \ \ \ \forall \alpha \in A

    implies that  x \in f^{-1} (E_ \alpha ) \ \ \ \ \ \forall \alpha \in A

    Therefore we have  x \in \bigcap _ { \alpha \in A } f^{-1} (E_ \alpha )

    On the other hand, pick  x \in \bigcap _ { \alpha \in A } f^ {-1} (E _ \alpha )

    implies that  x \in f^{-1} (E _ \alpha ) \ \ \ \ \ \forall \alpha \in A

    so  \exists y \in E_ \alpha such that  f(y) = x

    Note that we can find such y for every  \alpha \in A , therefore,  x \in f^{-1} ( \bigcap _ { \alpha \in A } E_ \alpha )

    I have doubts about the second last line of my proof, is it reasonable for me to state that? Thanks!
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  2. #2
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    \begin{gathered}<br />
   \hfill \\<br />
  x \in f^{ - 1} \left( {\bigcap\limits_n {E_n } } \right)\, \Rightarrow \,f(x) \in \bigcap\limits_n {E_n } \, \Rightarrow \,\left( {\forall n} \right)\left[ {f(x) \in E_n } \right]\, \Rightarrow \,\left( {\forall n} \right)\left[ {x \in f^{ - 1} \left( {E_n } \right)} \right] \hfill \\ <br />
\end{gathered}
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