Suppose we have definied chern classes $c_i$ for line bundles for all $i\geq 0$.
Let $E\to B$ be a rank $r$ vector bundle. By the splitting principle there exists a map $f:Y\to B$ such that $f^*E$ is a sum of line bundles $L_1\oplus\ldots\oplus L_r$ and $f^*:H^*(B)\to H^*(Y)$ is injective.
Does the splitting principle define chern classes for vector bundles if they are known for line bundles?
The splitting principle says only, that if the chern classes are defined in two ways (natural with respect to continuous maps), one can check on line bundles if the two ways end up to be the same. I don't see how the splitting principle could be used to define the classes because I fear that the decomposition into line bundles (or, the space $Y$ such that $f^*E$ splits) is not unique.