# Cropping some part of a tikzpicture

I have a code for two intersecting circles:

\begin{tikzpicture}
\draw (0,0) circle (1cm);
\draw (2,-2)  circle (2cm);
\end{tikzpicture}


which outputs this:

And I want to zoom into the intersection area without writing a new block of code. I can do this by using arcs:

\begin{tikzpicture}
\tikzstyle{intersection} = [draw, circle ,fill=darkgray, inner sep=0.3mm]
\draw (0,0) arc (-90:70:1cm);
\draw (2,1.5)  arc (100:200:2cm);
\end{tikzpicture}


which outputs this:

which resembles what I want. But it is very inconvenient if I do this over and over again. So, is there a simpler way to reuse a cropped part of a tikzpicture?

\documentclass[tikz, margin=3mm]{standalone}