Imagine a model with stock prices and dividends of these stocks, as well as a market bond with associated short rate process. It is known that this model is arbitrage-free if there exists an equivalent martingale measure $Q$.

It is then asserted that the price of a call option at time $t$ is the discounted conditional expectation under the equivalent martingale measure $Q$ of its payoff.

Question: Why is this true? The way I think about it is that if we imagined that the call option is a new stock which we introduce to the market, then we can consider it as a stock with no dividends up until expiration date where the final dividend is then its payoff. If we imagine "adding" this stock to our model, then it would remain arbitrage-free if and only if this "new stock" was priced so that $Q$ remains an equivalent martingale measure, and this precisely meanas that the call option price at time $t$ needs to be given as that conditional expectation (discounted).

Is this reasoning correct?