I am trying to see that Isotropy group/object group/vertex group of a Lie groupoid is a Lie group.

Let $\mathcal{G}$ be a Lie groupoid and $x$ be an object in $\mathcal{G}$ i.e., $x\in \mathcal{G}_0$. By an isotropy group of $\mathcal{G}$ we mean the collection of all arrows from $x$ to itself. Some people write $\mathcal{G}(x,x)$ or $\mathcal{G}_x$ for this set.

This is a group as we are in the case of groupoid where all arrows are invertible and as source and target are same for each map.

If we write down what actually is $\mathcal{G}_x$ is, we have $$\mathcal{G}_x=\{g:x\rightarrow x\}=s^{-1}(x)\cap t^{-1}(x)$$ where $s,t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ are source and target maps. Only extra information given on them is that they are smooth submersions. As $s:\mathcal{G}_1\rightarrow \mathcal{G}_0$ is a submersion, $s^{-1}(x)$ is an embedded submanifold of $\mathcal{G}_1$ and similarly as $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ is a submersion, $t^{-1}(x)$ is an embedded submanifold of $\mathcal{G}_1$. I am not able to see why would their intersection is an embedded submanifold. I guess it has something to do with transversal intersections.

Two maps $F,G:M\rightarrow N$ are said to intersect transversally if $$F_{*,a}(T_aM)+G_{*,b}(T_bM)= T_p(N)$$ where $a,b\in M$ such that $F(a)=G(b)=p$.

If atleast one of maps $F,G:M\rightarrow N$ is a submersion, then they intersect transversally.

As $F_{*,a}$ or $G_{*,b}$ is surjective, we already have $F_{*,a}(T_aM))=T_pN$ or $G_{*,b}(T_bM))=T_pN$, which means $$F_{*,a}(T_aM)+G_{*,b}(T_bM)= T_p(N).$$

In particular, for source and target maps $s,t$ as above, we have transversal intersection. Thus, by transversal intersection theorem. we see that $$\{(g,h):s(g)=t(h)\}\subset \mathcal{G}_1\times \mathcal{G}_1$$ is a smooth manifold.

But here what I am looking for is little extra.

The set $s^{-1}(x)\cap t^{-1}(x)$ is a subset of $\{(g,h):s(g)=t(h)\}=\mathcal{G}_1\times_{\mathcal{G}_0} \mathcal{G}_1$. How do I see that $\mathcal{G}_x=s^{-1}(x)\cap t^{-1}(x)$ is also a submanifold? How do I see that with this smooth structure, this is a Lie group?

Any suggestions are welcome. I am reading Orbifolds as Groupoids.