Prove that $\mathcal{M}=\{(x, y) \in X \times Y : f(x) = g(y)\}$ is a smooth submanifold.

Prove that $\mathcal{M}=\{(x, y) \in X \times Y : f(x) = g(y)\}$ is a
smooth submanifold.

Let $\mathcal{X}^m, \mathcal{Y}^n,\mathcal{Z}^p$ be manifolds. Let $f :
\mathcal{X} \to \mathcal{Z}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be maps
such that $f \pitchfork g$. Prove that $$\mathcal{M}=\{(x, y) \in X \times
Y : f(x) = g(y)\}$$ is a smooth submanifold of $\mathcal{X} \times
\mathcal{Y}$.



It seems what I have attempted is completely wrong...