"I am full but I am eating an egg."
Translation:Tokum ama yumurta yiyorum.
It's a general rule for verb stems that end in a vowel.
If it's one of the four vowel harmony vowels, you don't see a difference (e.g. "uyuyor" could be either "uyu- + -yor" or "uyu- minus u = uy- + -uyor") but, for example, "ödemek" becomes "ödüyor" and not "ödeyor, ödeiyor, ödiyor, ödeyiyor" or anything like that.
With "yemek", that would theoretically give "ye- minus final vowel = y- + İyor" with no vowel left to harmonise with :) But it harmonises with the stem vowel, so yemek: yiyor, demek: diyor.
However, "yemek, demek" are irregular in that they turn into yi-, di- before "y" in other cases (in particular in the future and the potential: yiyecek, yiyebilir) even though those don't normally "eat" the vowel and replace it with anothe rone.