While reading GMTW - The homotopy type of the cobordism category (https://arxiv.org/abs/math/0605249, p. 16) I found the following passage:

Lef $f:W\to \mathbb{R}$ be the projection. Then $(\pi_2,f):W\to X\times \mathbb{R}$ is proper since we have assumed that $X$ is compact. For $n\gg d$ we get an embedding $W\subset X\times \mathbb R\times \mathbb R^{d-1+n}$ which lifts $(\pi_2,f)$.

In this context I think $(\pi_2,f):W\to X\times \mathbb{R}$ is a submersion so by Ehresmann's lemma it is actually a fiber bundle. I'm not sure where the embedding comes from though. Is this some version of Whitney's theorem that is good for fiber bundles?