$V[G]=V[f_{G}]$ if $f_{G}=\bigcup\{s:(s,H)\in G\}$.

Question

Eventually different forcing, $\mathbb{E}$, consists of pairs $(s,F)$, where $s \in \omega^{<\omega}$ and $F$ is a finite set of reals.

$(s,F)\leq (t,G)$ iff $t \subseteq s$ and $G \subseteq F$ and $\forall{i \in \text{dom}(s\setminus t)}\forall{g \in G}(s(i)\neq g(i))$. It generically adds a new real $f_{G}=\bigcup\{s:(s,H)\in G\}$.

Question: How can I show that $V[G]=V[f_{G}]$ if $f_{G}=\bigcup\{s:(s,H)\in G\}$.

Any suggestion please.

Answer 1

HINT: Show that $G=\{(f_G\restriction n,F)\mid n\in\omega, F\text{ finite set of reals in }V\}$.