$M$ is a transitive proper class model of $ZFC$.
Let $T$ be a tree on $\omega \times \omega$ and $T \in M$. We know $proj([T])$ are precisely the $\Sigma^1_1$ sets (i.e, $\Sigma^1_1$ = $\omega$-suslin). Oh and by the way we know being a tree is absolute, $\omega$ is absolute, etc.
So we want to show $(x \in proj([T]))^M \iff x \in proj([T])$
$\implies$ is pretty easy.
$\impliedby$ is what I'm looking at.
I'm aware of the result that $\Pi^1_1$ sets are $\omega_1$-suslin, and that does a lot of the heavy lifting for me. I now have a tree $U$ on $\omega \times \omega_1$ such that $proj([U]) = \omega^\omega - proj([T])$.
Is this the right way to leverage this info to prove the result:
$x \in proj[T] \rightarrow x \not \in proj[U] \rightarrow (x \not \in proj[U])^M$ (this comes from $\implies $ direction).
Now I want to say $(proj[U] = \omega^\omega - proj[T])^M$ is $(proj[U])^M = (\omega^\omega)^M -(proj[T])^M$, and since $(\omega^\omega)^M = \omega^\omega \cap M$, we get that
$(x \not \in proj[U])^M \rightarrow (x \not \in \omega^\omega - proj[T])^M \rightarrow x \not \in \omega^\omega \cap M - (proj[T])^M \rightarrow x \in (proj [T])^M \rightarrow (x \in proj[T])^M$
Thus $x \in proj[T] \rightarrow (x \in proj[T])^M$ giving us $\impliedby$ direction.
Is this the right idea? And if not, where did I go wrong?
Here's a few roadblocks I can think of:
I made some relativization mistake in my mess of implications.
More importantly, I'm more worried about problems of absoluteness with $\omega_1$ and trees on $\omega_1$, and $\omega_1$-suslin sets. If this is indeed a problem I think I will have to try and use the fact that $x \in proj[T] \iff T_x$ is ill founded, to get around $\omega_1$ issues.
There's no reason to bring $\omega_1$, Suslinness, etc. into this. Instead, you should use the following:
Note that $M$ doesn't have to be a proper class! Similarly, Shoenfield absoluteness applies to set-sized structures as well: for Shoenfield, we merely need $M$ to be a transitive model of $\mathsf{ZFC}$ with $\omega_1^V$ as an element - which is more than in Mostowski, but still much less than proper classhood.
The proof is as follows. Clearly ill-foundedness in $M$ yields ill-foundedness in reality, since "is a descending sequence" is absolute. The other direction is the nontrivial one. Suppose $M$ thinks $P$ is well-founded. $\mathsf{ZFC}$ proves (and this is a good exercise) that a partial order $Q$ is well-founded iff it admits a ranking function, that is, a function $f:Q\rightarrow\alpha$ such that $\alpha$ is an ordinal and $a\le_Qb\implies f(a)\le f(b)$. Since $M\models\mathsf{ZFC}$ and $M$ thinks $P$ is well-founded, there is in $M$ some $f$ such that $M$ thinks $f$ is a ranking function for $P$. Being a ranking function is absolute (the key point being that being an ordinal is absolute), and so in reality $f$ is a ranking function. But the existence of a ranking function implies well-foundedness, so $P$ is well-founded in reality.
By considering $T_x$, Mostowski absoluteness gives the result you want - for arbitrary transitive $M\models\mathsf{ZFC}$, not just the proper class sized ones.
In fact, even that's overkill: all we needed from $M$ was transitivity and some very basic set theory, much less than $\mathsf{ZFC}$ - Mostowski absoluteness holds for all transitive models of $\mathsf{KP}$ + "Every set is contained in an admissible set."