This question requires a comprehensive answer, perhaps beyond the confines of my input box :) Suffices here to state the following:
The First Fundamental Theorem of Asset Pricing states that in an arbitrage-free market, there exists a ("net") present value function, that is, a linear valuation rule whose value is zero when evaluated in any traded cash-flow.
This is an existence theorem, and it does not depend on the theoretical or "real" form of the market. It does not depend on discrete or continuous time modeling, as it does not depend on whether there are transaction costs, trading constraints, or missing markets. All we need to have is the assumption that we can undertake two or more trades simultaneously, that we can scale them up, and that for every given trade, we can have its "mirror" in the market - that is, that we have a linear vector space of traded cashflows.
The Second Fundamental Theorem of Asset Pricing states that when an arbitrage-free market is "complete", the linear valuation rule is unique.
It is also true that these two separate theorems with different implications, are more often than not, presented in a fused form. This can be confusing. Proofs of these facts are virtually in every graduate asset pricing book. My favourite one is Duffie's 'Dynamic Asset Pricing Theory'.