Reference for bijection in distributive lattices

Question

I search a reference (in a book if possible) for the following for finite distributive lattice $L$ with minimum $m$ and maximum $M$. Let $M(L)$ be the set of meet-irreducibles of $L$ and $J(L)$ the set of join-irreducibles of $L$.

Then $\phi: M(L) \rightarrow J(L) $ given by $\phi(x)=\min ( L \setminus [m,x])$ is a bijection and $x$ is meet-irreducible iff $|\min ( L \setminus [m,x])|=1$

Answer 1

I found the result in the book by Steven Roman on lattices and orders as theorem 4.27.