We'll be fine. We'll have to make adjustments, but we'll be fine. So will the planet, and so will the vast majority of animals.
You know that the extinction rate of species (which includes animals) is running at approximately 200 per day.
Unless you mean the vast majority of animals in the sense of numbers of individual animals, instead of diversity of species, then maybe you're right: the vast majority will be fine as the biomass of domestic animals now outweight the biomass of every other species.
If we want to be alright, we'll probably want to start reducing the cattle headcount.
Humans aren't likely to die out directly, although scenarios where it all seems a bit pointless and the remaining few million humans stop reproducing are plausible.