I am trying to show that the definition of locally of finite type given in Hartshorne(ch. 2.3) is equivalent to the definition of locally of type P given in Stacks 29.14. One direction is clear: if a morphism $f: X \to Y$ is locally of finite type as in Hartshorne, then $f$ is locally of type P. Conversely, though, if $f$ is locally of type $P$, I do not see how we obtain that $f$ is locally of finite type.
Say $f: X \to Y$ is a morphism of schemes which is locally of type P, where P is defined in the title. Then, following the definition, we know that for every $x \in X$ there is an open affine neighborhood $U$ of $x$ and an affine open $V \subseteq Y$ so that $f(U) \subseteq V$ and the induced ring homomorphism $$\mathscr{O}_Y(V) \stackrel{f^{\#}}\to \mathscr{O}_X(f^{-1}(V)) \stackrel{\rho}\to \mathscr{O}_X(U)$$ is of finite type.
To show that $f$ is of finite type as in Hartshorne, we need an open cover of $Y$ by open affines. In the definition above, we can avoid fixing an element by taking an open cover $\{U_i\}$ of $X$ where $V_i$ are those open affine subsets of $Y$ where the composite $\mathscr{O}_Y (V_i) \to \mathscr{O}_X(U_i)$. The problem is that we need an open cover of $Y$ to satisfy the definition given in Hartshorne. I see no reason that these $\{V_i\}$ should cover $Y$ and no clear way to construct an open cover of $Y$ from these $\{V_i\}.$
It seems like we could just come to the right conclusion using Stacks Lemma 29.14.3 but it seems like this equivalence should be more elementary. Do any of you have any hints or ideas?
Thanks!