TITLE: Intersection of inverse images
QUESTION [5 upvotes]: Given $A$ and $B$ is the subset of $C$ and $f:C\mapsto D$,
$$f(A\cap B)\subseteq f(A) \cap f(B)$$ and the equality holds if the function is injective.
But why for the inverse, suppose that $E$ and $F$ is the subset of $D$,
$$f^{-1}(E \cap F) = f^{-1}(E) \cap f^{-1}(F)$$
without saying that the inverse function is injective. So if 
$$x\in f^{-1}(E) \cap f^{-1}(F)$$
$$x\in f^{-1}(E) \text{ and } x\in f^{-1}(F)$$
This means that there exists elements $y_1 \in E$ and $y_2 \in F$.
So here how do we know that these two elements are equal.
I am independent learner so I hope I can get an explaination in more details.

REPLY [3 votes]: proof: $$x\in f^{-1}(E \cap F)\Leftrightarrow f(x)\in E\cap F\\\Leftrightarrow f(x)\in E \  and \ f(x)\in F\\\Leftrightarrow x\in f^{-1}(E) \ and \ x\in f^{-1}(F) \\
\Leftrightarrow x\in f^{-1}(E)\cap f^{-1}(F)\\$$
so the equality holds.