Does a von Neumann universe $V$ satisfy the $\omega$-global covering property for $V$ itself?

Question

$V$ is an inner model of itself.(ZF(C): Model or Inner Model)
Therefore the following theorem, Bukovský's theorem is satisfied:

Let N ⊆ M be an inner model.
The following are equivalent:
1. N satisfies the $κ$-global covering property for M.
2. There is a $κ$-c.c partial order P ∈ N and a generic G ⊆ P such that M = N[G].
(ref. this)

In conclusion, V is an inner model of itself and there is ${\omega}$-c.c P and generic G such that V = V[G](trivial forcing extension) so V satisfies the ${\omega}$-global covering property for V itself.

Question: It is it correct?

Thanks in advance.

Answer 1

Sure, but it's also massive overkill. $V$ trivially satisfies the $\kappa$-global covering property: just take $F(x)=\{f(x)\}$. There's no reason to invoke Bukovsky's theorem. (Indeed, both properties - that $V$ is an $\omega$-c.c. generic extension of itself, and that $V$ has the $\omega$-global covering property - are basically trivial.)