Let $F\hookrightarrow E \overset{\pi}{\longrightarrow} M$, $F'\hookrightarrow E' \overset{\pi'}{\longrightarrow} M'$ two smooth fiber bundles.
If $F'\subset F$, $M'\subset M$, $E'\subset E$ are embedded submanifolds and $\pi'=\pi|_{E'}$. Then is the following chart condition true?
$\forall p \in M'$, $\exists U\ni p$ open in $M$ and a chart $\phi:\pi^{-1}(U)\longrightarrow U\times F$ of $E$ that $\phi|_{\pi'^{-1}(U\cap M')}:\pi'^{-1}(U\cap M')\longrightarrow (U\cap M') \times F'$ is a chart of $E'$
Ps: I have the later for definition of subbundle, but I think it's equivalent to the first.