Realm Transfer Question

Hello, I just started a Realm Transfer and it is my first time.

It says it may take a few days, but my characters realm got changed within the hour, the character has appeared in my new realm already. But I haven't gotten a confirmation e-mail yet. In my transaction section for the realm changing page, my transfer is under 'Successful'.

I'm just curious, is it safe to start playing on my character and is it truly finished? Or is it is necessary to wait longer, since I never got a confirmation e-mail and the transfer went through oddly quicker than expected.