Proof of $f^{-1}(B_{1}setminus B_{2}) = f^{-1}(B_{1})setminus f^{-1}(B_{2})$

I want to prove the following equation:

$$
f^{-1}(B_{1}setminus B_{2}) = f^{-1}(B_{1})setminus f^{-1}(B_{2})
$$

Is this a valid proof? I am not sure, because at one point I am looking at $f(x) in B_1$, but then $x in f^{-1}(B_1)$ could be actually some different points.

$$begin{align*}
x in f^{-1}(B_{1}setminus B_{2}) &iff f(x) in B_{1}setminus B_{2} \
&iff f(x) in B_{1} land f(x) notin B_{2} \
&iff x in f^{-1}(B_{1}) land x notin f^{-1}(B_{2}) \
&iff x in f^{-1}(B_{1})setminus f^{-1}(B_{2})
end{align*}$$