set theory - a question about diagonal prikry forcing

Suppose <kappa_n|n<omega> is a strictly increasing sequence of measurable cardinals,

kappa is the limit of this sequence. For each n<omega, U_n is a normal measure on

kappa_n. P is the diagonal Prikry forcing corresponding to kappa_n's and U_n's.
Suppose g is P-generic sequence over V. We have known that for each strictly increasing

sequence x of length omega such that each x(i)<kappa_i and xin{V}, x is eventually

dominated by g. In V[g], suppose A is a subset of kappa, A is not in V. Is there a strictly

increasing sequence y of length omega such that each y(i)<kappa_i and yin{V[A]}, y is not

eventually dominated by g?

(g can eventually dominate all such sequences in V, V[A] is greater than V, I feel g can not

eventually dominate all such sequences in V[A].)