Is this property about a monotonic function between DCPOs true?

Question

Let $P$ anq $Q$ be dcpos, $f : P \to Q$ be a monotonic function and $D$ be a directed subset of $P$.

Is it the case that: if $y$ is an upper bound of $f(D)$, then $f(\vee D) \le y$.

Answer 1

Counterexample: $P=Q=([-1,1],\le)$, $D=[-1,0)$, $f=\mathbf1_{[0,1]}$ (indicator function), $y=0$.