Reference for "It is enough to specify a sheaf on a basis"? The wikipedia article on sheaves says:

It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a basis for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering. Thus a sheaf can often be defined by giving its values on the open sets of a basis, and verifying the sheaf axioms relative to the basis.

However, it does not cite a specific reference for this statement. Does there exist a rigorous proof for this statement in the literature?

This is proven in Serres FAC, Chapter 1 Section 1 subsection 4. 
His definition of a sheaf is what is currently called an etale space and a modern pre sheaf is what Serre refers to as a system, then a modern sheaf is a system satisfying propositions 1 and 2. The categories of sheaves over X and etale spaces over X are equivalent though.
Edit:
I originally said subsection 3, but it is subsection 4.
