Let $V$ be vector space over $\mathbb{F}$, and $W\subseteq V$ a subspace. Let $p:V\rightarrow V/W$ be the canonical projection. Let $X$ be the set of all subspaces containing $W$ and $Y$ be the sets of all subspaces of $V/W$. I want to show that $p$ induces a bijection in the following way:
$L\in X $ is mapped to $p(L)=\{p(v) \mid v\in L\}$
$M \in Y$ is mapped to $p^{-1}(M)=\{v\in V \mid p(v) \in M\}$.
I feel as though I'm just having trouble with the set theory. Namely, I think I'm on the right track by just showing that each one of these inverts the other, but I'm having trouble sifting through the notation...
Help with what I just mentioned, and getting some intuition for what this map is telling me would be most appreciated!