I have trouble understanding a key step of Lemma 01ST on the Stack Project. The problem is showing that $U_u'=f^{-1}(V_u) \cap U_u$ is standard affine open in $U_u$, where $U_u\subset U$ itself is standard affine open. I will rewrite the problem so that unrelated details are skipped:
Let $f:X \to Y$ be a morphism of schemes. Let $U$ and $V$ be affine open subsets of $X$ and $Y$ respectively such that $f(U) \subset V$. For $u \in U$, there exists a standard affine open set $U_u \subset U$ and an affine open set $V_u \subset Y$. We can pick a standard affine open set $V_u' \subset V_u \cap V$. How do I show that $U_u'=f^{-1}(V_u') \cap U_u$ is standard affine open of $U_u$ and therefore $U$?
I understand this may be a "check the definition" problem, but I haven't figured out where should I go for. I believe assuming $U_u'$ is not standard affine open will quickly induces a contradiction, which I personally believe may come from the fact that standard affine open sets form a basis.
There is also a chance that I have missed some algebraic properties.
Any help will be appreciated.