$\textbf{Lemma: }$Let $U,V$ be open in $\mathbb{R}^n$, and let $\varphi:U\mapsto V$ be a diffeomorphism in $C^1$. Let $f:V\mapsto \mathbb{R}$ be Riemann integrable, with compact support contained in $V$, then $f\circ \varphi \lvert \det\varphi’\rvert$ is Riemann integrable in $U$, with compact support contained in $U$, and $$\int_V f(y)dy=\int_U f\circ \varphi(x) \lvert \det\varphi’(x)\rvert dx. $$
In my Analysis book, the proof of the given lemma was omitted, however, I am curious to know the reason behind it. Is there any reference that I could use to understand the above statement?
