Preimage simple problem Which one is correct and which one is wrong ?


*

*$f^{-1}[Y \cap Y^{'}] \subseteq f^{-1}[Y] \cap f^{-1}[Y^{'}]$

*$f^{-1}[Y] \cap f^{-1}[Y^{'}] \subseteq f^{-1}[Y \cap Y^{'}]$


Here is my solution:
\begin{align*}
x \in f^{-1}[Y \cap Y^{'}] &\ \ \Leftrightarrow  f(x) \in Y \cap Y^{'} \\
&\ \ \Leftrightarrow \ \ f(x) \in Y \wedge f(x) \in Y^{'} \\
&\ \  \Leftrightarrow \ \ x \in f^{-1}[Y] \wedge x \in f^{-1}[Y^{'}]\\
&\ \  \Leftrightarrow \ \ x \in f^{-1}[Y] \cap f^{-1}[Y^{'}]
\end{align*}
is it right? can we conclusion $x \in f^{-1}[Y] \wedge x \in f^{-1}[Y^{'}]$ from $f(x) \in Y \wedge f(x) \in Y^{'}$ ? I need proof for this part if my solution is correct. thanks

Your series of bidirectional implications are correct and prove that both statements are true.
