Does anyone know of a reference for one or both of the following facts (in $\mathsf{ZF}$)?
- If the set of reals $A$ is Suslin, then every $\Sigma^1_1(A)$ set of reals is Suslin.
- If $T$ is a tree on $\omega \times \kappa$ and $V[G]$ is a generic extension of $V$, then a $\Sigma^1_1(\text{p}[T])$ statement holds in $V$ if and only if the corresponding $\Sigma^1_1(\text{p}[T]^{V[G]})$ statement holds in $V[G]$.
Here a set of reals is called Suslin if it the projection of a tree on $\omega \times \kappa$ for some ordinal $\kappa$, and (from an answer by Carlo Von Schnitzel on Math.SE) we say that a set of reals $B$ is $\Sigma^1_1(A)$ iff there is a $\Sigma^1_1$ set of reals $D$ such that $$B(x) \iff \exists y\, \big(\forall n\, (y)_n\in A \wedge D(x,y)\big).$$ (I'm oversimplifying a bit here; in the special case that $A = \emptyset$ we need to define $\Sigma^1_1(A) = \Sigma^1_1$.)
It seems like these facts are often used in an informal way (e.g. "consider the tree $T'$ of attempts to build reals coding countable objects $x$, $y$, and $z$ having some property that is witnessed by a branch of $T$....")
It would be nice to have a reference for a general statement that encompasses all such arguments.