= Solution
Let $a,p\in M[G]$, and suppose a <first-order formula> $\psi(x,y,p)$ defines exactly one $y$ for every $x\in a$. Apply the <Lévy reflection theorem> to the formulas needed to express this assertion, choosing an <ordinal> $\alpha$ with $a,p\in V_\alpha^{M[G]}$ such that
$$
M[G]\models\exists y\,\psi(x,y,p)
\quad\Longleftrightarrow\quad
V_\alpha^{M[G]}\models\exists y\,\psi(x,y,p)
$$
for every $x\in a$. Therefore every required value lies in the <set> $V_\alpha^{M[G]}$.
The already established <axiom schema of separation> forms the set
$$
b=\{y\in V_\alpha^{M[G]}:\exists x\in a\,\psi(x,y,p)\}.
$$
Functionality makes $b$ exactly the range of the definable function on $a$. This proves every instance of the <Axiom schema of replacement> in the <generic extension> $M[G]$.
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