# Operator K-theory and Topological K-theory

I was trying to understand the relationship between these two K-theory's when you pick your C* algebra to be $$C(X)$$ for $$X$$ a compact Hausdorff space. For this you create a function between $$P_\infty (C(X))$$ and the Vector bundles on $$X$$ by $$p$$ goes to $$\xi_p=(E_p,\pi,X)$$ where $$E_p =\{(x,v) \in X\times \mathbb{C}^n: v\in p(x)(\mathbb{C}^n)\}$$ and $$\pi$$ is the natural projection. I wonder why this is well defined in the equivalence classes of $$P_\infty (C(X))$$. i.e why $$p\sim q$$ $$\iff$$ there exists an isomorphism of vector bundles between $$\xi_p$$ and $$\xi_q$$.