Disconnected components in graphical models

Suppose we have a graphical model defined by a directed acyclic graph $$G$$. Suppose that a node $$a$$ divides $$G$$ into two connected components. By this I mean that:

$$G=G_1 \cup G_2 \cup \{a\}$$

$$G_1 \cap G_2 = \emptyset$$

and there are no edges between $$G_1$$ and $$G_2$$. ( ok technically by $$G$$ I mean the set of vertices of $$G$$, I hope the question is clear anyway... )

Is it true that, given $$g_1 \in G_1$$ and $$g_2 \in G_2$$:

$$p(g_1|g_2,a)=p(g_1|a) ? [1]$$

I conjecture this from how graphical models are build, but is it true ? Of course it is simple to generalize [1] but I wanted to write a simple statement.

For example if you follow the d-separation procedure, at step 4, you will delete $$a$$ which necessarily disconnects $$g_1$$ and $$g_2$$, $$\forall g_1\in\mathcal{G}_1$$, $$\forall g_2\in\mathcal{G}_2$$, leading to the conditional independence.