Lemma 3.27. Assuming that:

$M ⊨ ℙ$ has countable chain condition
$X, Y \in M$
$G$ is $ℙ$ -generic over $M$
$f : X \to Y$ , $f \in M [G]$

Then there is

F \in M

such that

\forall x \in X, F (x) \subseteq Y

,

\forall x \in X, f (x) \in F (x)

and

M ⊨ \forall x \in X, F (x) is countable

.