As the conditions for transformation of multiple integrals, many textbooks state two separate conditions (along with other conditions): (1) the transformation is 1-1 (2) The Jacobian does not vanish
For example, Courant and John in "Introduction to Calculus and Analysis" state:
If the transformation $x=\phi(u,v), y=\psi(u,v)$ represents a continuous 1-1 mapping of the closed Jordan-measurable region $R$ of the $x,y$-plane on a region $R'$ of the $u,v$-plane, and if the functions $\phi$ and $\psi$ have continuous first derivatives and their Jacobian $$\frac{d(x,y)}{d(u,v)}=\phi_u \psi_v-\psi_u\phi_v$$ is everywhere different from zero, then $$ \iint_R f(x,y)dxdy=\iint_R f(\phi(u,v),\psi(u,v))\left|\frac{d(x,y)}{d(u,v)}\right|du dv > $$
or Zill and Wright in "Advanced Engineering Mathematics":
The functions f and g have continuous first partial derivatives on S.
The transformation is one-to-one.
Each of the regions R and S consists of a piecewise-smooth simple closed curve and its iterior
The determinant $$ \begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v}\\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{vmatrix} $$ is not zero on $S$.
If the transformation is $C^1$, doesn't "$J\neq 0$ within the interior of the preimage region" imply that the transformation is 1-1? or doesn't being 1-1 imply that $J\neq 0$?
Why are they stating these as two separate conditions?
PS: Bartle in "Elements of Real Analysis" states the theorem as follows:
Suppose $\phi$ is in Class $C'$ on an open subset $G$ of $\mathbb{R}^p$ with values in $\mathbb{R}^p$ and that the Jacobian $J_\phi$ does not vanish on $G$. If $D$ is a compact subset of $G$ which has content and if $f$ is continuous on $\phi(D)$ to $R$, then $\phi(D)$ has content and $$\int_{\phi(D)} f=\int_D (f\circ\phi)|J_\phi|.$$
As we can see here Bartle does not talk about the fact that $\phi$ is a one-to-one onto mapping.