Proving that $\Delta(M \times M)$ is a submanifold of $M \times M$ I am struggling to prove that

$\Delta(M \times M) = \{(x,x) : x \in M\}$ is a submanifold of $M \times M$.

A manifold M is a submanifold of N if there is an inclusion map $i:M \rightarrow N$
such that:

*

*$i$ is smooth


*$Di_x$ is injective for each $x \in M$


*The manifold topology of $M$ is the induced topology from $N$.
I also know the following theorem:

Let $F:M \rightarrow N$ be a smooth map and $c \in N$ be such that at each point $a \in F^{-1}(c)$ the derivative $DF_a$ is surjective.
Then, $F^{-1}(c)$ is a smooth manifold of dimension $dim(M)-dim(N)$.

In the course of the proof, we see that the manifold structure on $F^{-1}(c)$ satisfies the conditions of the definition of submanifolds.

Problem:

We want to avoid the verifications of the definition of submanifold.
Applying the theorem, we can take $F:M \times M \rightarrow M$ defined by $(x,y) \mapsto y-x$, and so $\Delta(M \times M) = F^{-1}(0)$.
But $y-x$ in general may not belong to $M$.
What do I need to do to prove what I want to prove? Hints? Thanks

Hint: Consider the map $M \to M \times M$ given by $x \mapsto (x,x)$.
