quantum mechanics - Part of a Wigner theorem

I was trying to understand why there should exist operator in Hilbert space to correspond to any symmetry transformation and found about Wigner's theorem. In it, I can see that any transformed vector in Hilbert space can be represented as a linear combination of transformed base vectors leading to something like, action of any operator on the specific state can be expressed as the action of this operator on the base states. How this proves existence of the operator I do not see. It seems to me that this operators is linear or antilinear but this is given it actually exists. Am I missing something? Can anyone point out part of the theorem which proves pure existence and not the properties of the operator? So to add something, symmetry only tells us ho a ray is transformed into another ray, it does not tell us which of the vectors of the old ray went into which of the new one. Where in the Wigner theorem can we see proven that such a connection can be established?