I want to use splitting principle to construct Stiefel Whitney classes assuming we know Stiefel Whitney class for a line bundle. Splitting principle says that
Theorem: Suppose we have a vector bundle $\xi$ of rank $n$ over X. Then there is a space $Y$ and a map $f: Y \to X$ such that $f^*\xi$ is a direct sum of line bundles and $f^*: H^*X \to H^*Y$ is an injection.
So now we want to define $w(\xi)$ but we can only define $w(f^*\xi)=w(L_1)...w(L_n)$. Now if we knew this was in the image of $f^*$ we could say $w(\xi)$ is the unique element which maps to this. But splitting principle does not say the map is surjective. So my question is how does one show it is in the image?
My try: We want to show that $w(f^*\xi) \in Im(f^*)$. But we do secretly know that the Steifel Whitney classes exist so we should be able to prove that $w(L_1) \in Im(f^*)$. That means that there is a line bundle on $X$ which pulls back to this line bundle under $f$. Here I have used that line bundles are classified by $H^1(X,\mathbb{Z}/2)$. Now I am not able to prove this. I tried to see if this was clear from the construction of $Y$ via projectivisation but I could not prove it. Any hints or ideas are appreciated. Thank you.