= Solution
Fix a <logical formula> $\theta(x,y,\vec p)$ with parameters in $V$, and a <set> $a\in V$. Suppose $\langle V,\in\rangle$ satisfies that each $x\in a$ has a unique $y$ with $\theta(x,y,\vec p)$. Relativize every unbounded <quantifier> in $\theta$ to the definable <class in set theory> $V$, obtaining an ambient <first-order formula> $\theta^V$.
Transitivity of $V$ ensures that every member of $a$ belongs to $V$. The ambient relation
$$
y\in V\ \wedge\ \theta^V(x,y,\vec p)
$$
therefore defines a unique ambient <set> $y$ for every $x\in a$. Apply ambient <replacement> to obtain its range <set> $b$. By construction every element of $b$ belongs to $V$, so part (ii) gives $b\in V$.
Membership in this <set> $b$ is exactly the range relation required by <replacement> inside the structure $\langle V,\in\rangle$. Since the <logical formula> and parameters were arbitrary, \b[every instance of the <replacement> scheme holds in the hierarchy <class in set theory>]. The proof uses <formula relativization to a class> and the <closure> property from part (ii), not the unsupported assertion that $V$ is the whole ambient universe.
Back to article page
