Proof of intersection for preimage

Question

A book I am reading asked me to prove

$$F^{-1}(A \cap B) = F^{-1}(A) \cap F^{-1}(B) $$

where $A$ and $B$ are images of $F$

My attempt:

$$ F^{-1}(A) = \{x \space | \space F(x) \in A \} \\ F^{-1}(B) = \{x \space | \space F(x) \in B \} \\ \text{Let } C = A \cap B \\ F^{-1}(C) = \{x \space | \space F(x) \in C \} \\ F^{-1}(A \cap B) = \{x \space | \space F(x) \in A \cap B \} = F^{-1}(A) \cap F^{-1}(B)\\$$

Answer 1

I think that's fine, though I would have introduced the intermediate step

$\{x\mid F(x)\in A\cap B\}=\{x\mid F(x)\in A\}\cap \{x\mid F(x)\in B\}$.

Answer 2

x in $f^{-1}$(A $\cap$ B) iff f(x) in A $\cap$ B
iff f(x) in A and f(x) in B
iff x in $f^{-1}$(A) and x in $f^{-1}$(B)
iff x in $f^{-1}$(A) $\cap$ $f^{-1}$(B)

A and B are not images of f, whatever that could mean.
They are subsets of the codomain of f.