Proof that Namba Forcing preserves stationary subsets of $\omega_1$

I am looking for a proof (or reference) of the fact that Namba Forcing preserves stationary subsets of $$\omega_1$$. This fact is stated and used throughout the literature(whenever you google it you'll find examples), though no one is refering to a proof or giving one.

I will state a few relevant definitions regarding Namba forcing.

With Succ$$_T(t)$$ we denote the set of all immediate successors of $$t$$ in the tree, ie Succ$$_T(t)=\{t^\frown \alpha \ | \ t^\frown \alpha\in T \}$$

With trunk$$(T)$$ we denote the initial segment of the tree where the nodes didn't start to branch off, ie $$t$$ = trunk$$(T)$$ iff $$\forall s\in T : \ s\upharpoonright |t|=t$$.

Namba Forcing has the following poset $$P = \{T\subset \omega_2^{<\omega_1} \ | \ T$$ is a tree and $$\forall t\in T \ t$$ has an extension $$s$$ such that $$|Succ_T(s)|=\aleph_2 \}$$

$$P$$ is ordered by inclusion: $$T\leq S$$ iff $$T\subseteq S$$, ie smaller trees are stronger.

So far, I have only found some unclear hints towards a proof I am looking for: In Proper Forcing - Shelah chapter XII, a class of forcings $$UP_0$$ is defined, and state that it preserves stationary subsets of $$\omega_1$$. See Claim 3.2 4) page 407. At the top of page 395 it states that Namba forcing belongs to this class. Though no proof is given and the book refers to a paper by Shelah, unclear which one.

The $$$$A2 Theorem'' on page 86 is what you are looking for.
• Thanks! I have worked may way trough the proof and found one mistake, which is easy to fix. For claim (f), one defines r prime such that not only $\zeta\in\Omega_{r,g,\sigma}$ but in $\Omega_{r,g,\sigma\upharpoonright n}$ for $n$ such that $k<n\leq$#$\sigma$. – T. Jacobs May 29 at 15:06