First let me say I am aware of the other threads on this result. The reason for me making this thread is to find out whether or not my proof/proof attempt is correct.
The problem stated in full detail is given below.
Let $X,Y$ be sets and $f:X \rightarrow Y$, let $C,D \subseteq X$ and let $A,B \subseteq Y$. Prove that $f^{-1}(A) \cap f^{-1}(B)= f^{-1}(A \cap B)$.
Here is my attempt:
$f^{-1}(A) \cap f^{-1}(B)= { \{x\in X \mid f(x) \in A\}}\cap { \{x\in X \mid f(x) \in B\}}= \{x\in X \mid f(x) \in A\wedge f(x) \in B\}= \{x\in X \mid f(x) \in A\cap B \}=f^{-1}(A \cap B)$
If anyone would be kind enough to explain to me where I've gone wrong or made an unjustified assumption I would be very grateful!
P.S. Sorry about the bad formating
Your proof seems perfectly OK to me.
If you aren't sure because the set notation is confusing you, maybe rewriting it without so many sets would also work:
$$\begin{align}x\in f^{-1}(A)\cap f^{-1}(B)\iff &(x\in f^{-1}(A)\land x\in f^{-1}(B)\\\iff& f(x)\in A\land f(x)\in B\\\iff& f(x)\in A\cap B\\\iff& x\in f^{-1}(A\cap B)\end{align}$$