I'd like to find a reference for the following fact.

First, some background: we can define de Rham cohomology of a smooth manifold $X$ of dimension $d$ using the de Rham complex $$ \Omega^0_X\to \Omega^1_X \to\ldots \to \Omega^d_X $$ of smooth forms. We can equally define it using the distributional de Rham complex $$ D^0_X\to D^1_X\to \ldots \to D^d_X $$ where $D^i$ are the $i$-currents, i.e. linear functionals on $(d-i)$-forms with compact support. Now, suppose we have an inclusion $i:Z\to X$ of a closed submanifold of codimension $r$. Then there is a map of complexes $$ \Omega^i_Z \to \ker(D^{i+r}_X \to D^{i+r}_{X-Z}) $$ given by sending an $i$-form $\eta$ to the functional on $(d-r-i)$-forms $$ \omega \mapsto \int_Z \eta \wedge \omega. $$ This should induce the Gysin/excision isomorphism $H^i(Z) \cong H^{i+r}_Z(X)$ (alternately notated $H^{i+r}(X,X-Z)$). I think I know how to prove this, by using a comparison with singular cohomology and developing the theory of cycle classes, then using Poincare duality, but it's quite unwieldy and requires introducing a lot of auxiliary theory and constructions.

My **question** is: is there a more direct way to prove that the above map of complexes is a quasi-isomorphism? Or does someone have a reference where any proof (not necessarily direct) is given?