Proving the image of a convex polyhedron under a linear map is a polyhedronprove for...

Proving the image of a convex polyhedron under a linear map is a polyhedron
prove for $A\in {\mathbb{R}}^{m\times n}$ and a convex polyhedron $Q\subseteq {\mathbb{R}}^n$ that the set
is also a convex polyhedron. However, I am asked to do so using the following statement about convex polyhedra (which is easy to prove):
$P\subseteq {\mathbb{R}}^{m+n}\text{ is a polyhedron }⟹\{x\in {\mathbb{R}}^n\mid (x,y)\in P\text{ for some }y\in {\mathbb{R}}^m\}.(1)$
This seems like it should be easy but I'm having trouble.
One approach is to write $Q$ as a set of linear inequalities
$Q=\{x\in {\mathbb{R}}^n\mid Bx\le b\}$
and then try to write $A(Q)$ as as system of linear inequalities
$A(Q)=\{y\in {\mathbb{R}}^m\mid B{A}^{-1}y\le b\}.$
This doesn't quite work since A may not be invertible. More importantly, it does not use the statement (#1) given above.