Preserving powersets and preserving MA

Question

I am reading a proof that seems to imply that if we start with a model of $\mathfrak{c} = \aleph_2 $ and $MA_{\aleph_1} $, and if a forcing preserves $\mathcal{P}(\omega_1)$, that is, it doesn't add any subsets of $\omega_1$, then it preserves the value of the continuum and Martin's Axiom. Is that true in general? What if we start with a model where the continuum is larger and we still have the entire Martin's Axiom?

Answer 1

Over $\mathsf{ZFC}$, Martin's Axiom is equivalent to its restriction to posets of size $<\mathfrak{c}$. This is Lemma $16.12$ in Jech. In case $\mathfrak{c}=\aleph_2$, this is a consequence of not adding any subsets of $\omega_1$.