Let $M$ be a countable transitive model of ZF-P, $\mathbb P$ the set of injections from a countable subset of $\mathbb R$ into $\omega_1$ with $\le=\supseteq$. Let $G$ be a $\mathbb P^M$-generic filter over $M$, and consider the forcing extension $M[G]$. Then $f=\bigcup G\in M[G]$ is a bijection between $\mathbb R^M=\mathbb R^{M[G]}$ and $\omega_1^M=\omega_1^{M[G]}$. My questions are

(1) is the $f$-image of a closed interval in $M$, i.e., is $f([a,b]^M)\in M$?

(2) if (1) not true, is there at least a superset $S([a,b])\in M$ of $f([a,b]^M)$ such at $S(I)\cap S(J)=\emptyset$, if $I,J$ are disjoint closed intervals?

(3) if both (1), (2) are not true, is there some other forcing notion for which (1) or (2) is true?