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?