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:

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