Range of elliptic operator in sobolev space $W^{1,2}$

Question

my book claims that the weak elliptic operator $A:W^{1,2}(\Omega) \to W^{1,2}(\Omega)^\prime$ where, $\Omega$ is an open bounded region for example, $A(u)(v)=\int_\Omega \sum_{i,j=1}^n a_{ij} \ \partial_i v \ \partial_ju \ dx$ and $a_{ij}$ are symmetric functions in $L^\infty$, has the range $R(A)=\{ F \in W^{1,2}(\Omega)^\prime ; F(1)=0\}$. This is used to show that this operator has closed range with codimension 1, and thus is a fredholm operator, because the nullspace are the constant functions. That constant functions are mapped to zero is obvious, but i cant see why the converse is also true. Could someone help me out?