uniqueness of representation of spherical coordinates

Question

Let $\varphi: \left(0,\infty\right) \times [0,2\pi) \times [0,\pi) \rightarrow \mathbb{R}^{3}$ defined by the formula: $$\varphi(r,\theta,\psi)=(r\sin(\psi)\cos(\theta),r\sin(\psi)\sin(\theta), r\cos(\psi))$$ For any point $P \in \mathbb{R}^{3} \backslash \left\{ (0,0,z): z \in \mathbb{R} \right\}$, show that there is a unique triplet $(r,\theta,\psi)$ such that $\varphi(r,\theta,\psi)=P$.
so I denoted $P=(x,y,z)$ and found that: $$r=\sqrt{x^{2}+y^{2}+z^{2}}$$ $$\theta=\arctan(\dfrac{y}{x})$$ $$\psi = \arccos(\dfrac{z}{\sqrt{x^{2}+y^{2}+z^{2}}})$$ I'm stuck on proving the uniqueness of this representation. I tried to proof by contradiction that there are $(r,\theta,\psi) \neq (r_1,\theta_1,\psi_1)$ but couldn't get to the contradiction. Any help would be great.