TITLE: Bad subforcings of nice forcing notions
QUESTION [5 upvotes]: Let $\mathbb{P}$ and $\mathbb{Q}$ be two forcing notions. Recall that we say $\mathbb{Q}$ is a subforcing of $\mathbb{P}$ if there exists a regular embedding $\mathbb{Q} \to \text{r.o.}(\mathbb{P}).$
Question. Let $\mathbb{P} \in \{Add(\omega, \kappa), Col(\omega_1, \kappa) \}.$
(1) Is there a subforcing of $\mathbb{P}$ which is not Proper?
(2) Is there a subforcing of $\mathbb{P}$ which is not semi-Proper?
Remark. In Subalgebras of Cohen algebras need not be Cohen, Koppelberg and Shelah show that there is a subalgebra of Cohen forcing which is not Cohen.

REPLY [8 votes]: No.  A subforcing of a c.c.c. forcing is c.c.c.  A subforcing of a countably closed forcing is countably-strategically-closed, which implies proper.  (This is easy to see via countable elementary submodels.  Use the strategy to construct a generic condition.)
Furthermore, every subforcing of a proper forcing is proper.  Properness is equivalent to preserving stationary subsets of $[\kappa]^\omega$ for all $\kappa$.  The stationarity of $X \subseteq [ \kappa ]^\omega$ cannot be restored once killed, since killing it is just adding some $f : [\kappa]^{<\omega} \to \kappa$ such that no $x \in X$ is closed under $f$.  (See Abraham's chapter of the Handbook.)