Yes, the Bible states this in the story of Genesis after God creates Adam and Eve and the Garden of Eden. Although I myself believe in the theory of evolution, there are some vegetarians and animal rights
activists who do believe in the literal interpretation of the Bible. These individuals would point out that according to the literal interpretation of Genesis, no animal ever died in the Garden of Eden. Therefore, in the Garden of Eden, Adam and Eve and all the animals were vegetarians.
In this context, the phrase "dominion over the animals" could not have referred to killing and eating them. Many would argue that "dominion" is more accurately interpreted as "stewardship" rather than tyranny. Furthermore, since the Garden of Eden represents the highest ethical ideal, some people who believe in the literal interpretation of the Bible interpret this to mean that vegetarianism is the highest ethical ideal.
Quote from Animal Rights
and Vegetarian Ethics
by Eugene Khutoryansky