Suppose we have a smooth vector bundle $\pi: E \rightarrow B$ and two sub vector bundles $\pi_1: E_1 \rightarrow B_1$ and $\pi_2: E_2 \rightarrow B_2$ such that the bases $B_1$ and $B_2$ are submanifolds of $B$. Now suppose we would like to intersect both subbundles, that is we would like to define 'something like' $\pi_{1,2}: E_{12} \rightarrow B_1 \cap B_2$ where:

1.) $B_1 \cap B_2$ is a (smooth) submanifold of $B$.

2.) For each $b \in B_1 \cap B_2$ we define $\pi_{1,2}^{-1}(b):= \pi_{1}^{-1}(b) \cap \pi_{2}^{-1}(b)$ as the (standard set theoretic) intersection of the fibers of $\pi_1$ and $\pi_2$ and $E_{12}:=\bigcup_{b \in B_1 \cap B_2}\pi_{1,2}^{-1}(b)$.

3.) The dimension $dim(\pi_{1,2}^{-1}(b))$ is constant for all $b \in B_1 \cap B_2$.

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Is this a vector bundle?

Is it a sub(vector) bundle of $\pi$?

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P.S.: I know the discussion here:

About the intersection of two vector bundles

but since it doesn't solve the problem, I think its o.k. to ask my question anyway.