Monotonicity of non-goodness for Erdős 624: $H(n)>m$ and $N\ge n$ imply $H(N)>m$

Let $H(n)$ be the least $m$ such that some $f:\mathcal P(X)\to X$, $|X|=n$, satisfies $\{f(A):A\subseteq Y\}=X$ for all $Y\subseteq X$ with $|Y|\ge m$ (Erdős Problem 624). Theorem. If $1\le n\le N$ and $H(n)>m$, then $H(N)>m$.

An $m$-good map on an $N$-set restricts to an $n$-subset $U$ and, composed with a retraction $X\to U$, gives an $m$-good map on $U$.