TITLE: Theorem of Bukovsky characterizing ground models
QUESTION [8 upvotes]: It was mentioned in a talk that Bukovsky proved the following are equivalent for inner models $M \subseteq V$:
(1) There is a partial order $\mathbb P \in M$ and a $\mathbb P$-generic filter $G \in V$ over $M$ such that $V = M[G]$.
(2) There is a cardinal $\kappa$ such that for every ordinal $\alpha$ and every function $f : \alpha \to \mathrm{Ord}$ in $V$, there is $F : \alpha \to [\mathrm{Ord}]^{<\kappa}$ in $M$ such that $(\forall \beta < \alpha) f(\beta) \in F(\beta)$.
Is this written up somewhere?  If not, can you prove it?  (Obviously (2) $\rightarrow$ (1) is the hard part.)

REPLY [4 votes]: The original paper is ``Characterization of generic extensions of models of set theory''
A proof can be found in the master thesis of Giorgio Audrito,  "Characterizations of set generic extensions", which is available at Viale's home page here.
Also the paper ''On the set-generic multivers'' by Friedman-Fuchino-Sakai gives a proof. See Friedman's home page.
I may mention that $\kappa$ can be chosen to be the cardinal for which the forcing notion is $\kappa$-c.c.