If $f:X\to Y$, and $A\subseteq X$, $B\subseteq Y$, then the equation $f[A] \cap B\subseteq f[A\cap f^{-1}[B]]$ holds. Indeed, let $y\in f[A]\cap B$, then $y=f(x)$ for some $x\in A$; since $f(x)\in B$ we also have $x \in f^{-1}[B]$ and hence $x\in A\cap f^{-1}[B]$, and $y\in f[A\cap f^{-1}[B]]$.
I would like to generalize this proof to bounded lattices. Here the forward and preimage maps $f$ and $f^{-1}$ are just monotone maps with a Galois connection $f(a)\le b\iff a\le f^{-1}(b)$. Is the theorem $f(a)\wedge b\le f(a\wedge f^{-1}(b))$ still true? If not, what extra property do sets bring to the table here?