Let
$f: X\to Y$ be a quasi compact separated morphism
$\{U_i\}$ be an open affine covering of $X$
$V$ be an open affine subset of $Y$
In Hartshorne's AG book chapter III, proposition 8.7 uses ;
$\{U_i \cap f^{-1}(V)\}$ forms an open affine cover of $f^{-1}(V)$
Is it really true?