Choosing $P$-names.

Question

let $M$ be a $c.t.m$. let $P$$∈$$M$ let $G$$⊆$$P$ be $M$-generic. let $α$ be an ordinal. let $f$$∈$$M[G]$ with dom($f$)=$α$. must there exist a $F∈M$ with dom($F$)=$α$ s.t for every $β<α$, $F(β)$ is a $P$-name of $f(β)$ $?$ Thanks.

Answer 1

There is a very general principle in forcing that says:

Whenever $\Vdash \exists x \varphi(x,\tau)$ then there is a name $\dot{x}$ so that $\Vdash \varphi(\dot{x},\tau)$. So using that $\Vdash \exists x (\dot{f}(\alpha) = x)$ just find a name $\dot{x}_\alpha$ so that $\Vdash \dot{f}(\alpha) = \dot{x}_\alpha$. Let $F(\alpha) = \dot{x}_\alpha$.