Whenever a large volume of water goes down my kitchen sink drain, it causes a back up in my basement sink, with food particles coming up as well. The basement sink is located directly beneath my kitchen sink. Based on some advice posted in similar threads, I decided to the snake the line between my kitchen drain and my basement drain by putting the snake through the p-trap in the basement and working my way up. The snake was able to pull out a bit of food, but not a large amount, a just a few blobs of mush. I was hoping I had solved the issue, but after running water in my kitchen sink, the water started backing up again.

Here are some pics of the drain in the basement.

As you will see the basement drain is connected to some sort of big drain pipe which the kitchen drain also connects to. I'm not sure what the "big drain pipe" is, is it a main drain and could it be the source of the backup? Any help is appreciated but I am thinking about hiring a pro at this point.

Thanks in advance!