Reference request: Pullbacks along submersions are submanifolds and the induced map is a submersion

Question

I'm looking for an introductory book in differential topology in which there are proofs of the following facts:

1. Let $X,X',Y$ be smooth manifolds, $X\rightarrow Y$ a smooth map and $X'\rightarrow Y$ a submersion, then the pullback $X\times_Y X'$ is a smooth submanifold of $X\times X'$ and the induced map $X\times_Y X'\rightarrow X$ a submersion.
2. Let $f\colon X\rightarrow X'$ be a smooth maps between smooth manifolds and $Z\subseteq X'$ be a submanifold transverse to it, then the tangent space of $Z$ and its inverse image (which is a submanifold) are related by $$T_m(f^{-1}(Z))=(d_mf)^{-1}(T_{f(m)}Z)$$

I would be especially happy if the case where $X,X'$ and $Y$ may have boundary is treated as well.

I know how to prove it. I just need a reference to a textbook.