Verifying a bijection 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!
 A: In $V$ the elements of $V/W$ are subsets of $V$; in $V/W$ they’re points. Thus, the quotient is really just formed by chopping $V$ up into (nice) chunks and wadding each chunk up to make a single point. The map $p$ then sends each point of $V$ to the chunk containing it. Here’s a quick and dirty picture that may help you to visualize more easily what’s going on:

If $V$ is on the blackboard, $p$ is basically just squashing everything down into the chalk tray and collecting each coset of $W$ into a single lump of chalk dust.
A: Let $L \in X$.
We claim that $L = p^{-1}p(L)$.
Clearly $L \subset p^{-1}p(L)$.
Let $x \in p^{-1}p(L)$.
Then $p(x) \in p(L)$.
Then $p(x) = p(y)$, where $y \in L$.
Hence $x - y \in p^{-1}(0) = W$.
Hence $x \in L + W = L$.
Hence $p^{-1}p(L) \subset L$.
Let $M \in Y$.
We claim that $M = pp^{-1}(M)$.
Clearly $pp^{-1}(M) \subset M$.
Let $y \in M$.
Since $p$ is surjective, there exists $x \in V$ such that $p(x) = y$.
Then $x \in p^{-1}(M)$.
Hence $y \in pp^{-1}(M)$.
Hence $M \subset pp^{-1}(M)$.
