Alexandrov's mapping lemma

Question

I'm looking for a complete proof of Alexandrov's mapping lemma. I'd also like to have the intuition for it explained if that's at all possible. Or alternatively any pointers in the right direction would also be appreciated!

Here is the statement which I found at http://www.math.utah.edu/~treiberg/PolyhedraSlides.pdf

Suppose $\phi$ : A $\rightarrow$ B is a mapping between n dimensional manifolds that satisfy the following conditions:

1. Every component of B contains image points of A.
2. $\phi$ is one-to-one
3. $\phi$ is continuous
4. $\phi$ has a closed graph: if $\left\{b_{j}\right\} \subset B$ is a sequence consisting of image points $b_{j} = \phi(a_{j})$ for some $a_{j} \in A$ which converges $b_{j} \rightarrow b \in B$ as $j-> \infty$, then there exists $a \in A$ with $\phi(a) = b$ and a subsequence $a_{i_{m}}$ which converges to $a$ as $m \rightarrow \infty$

Then $\phi$ is onto, i.e., $\phi(a) = b$

Answer 1

A complete proof is given by Aleksandrov in his "Convex Polyhedra" book.