Does the random sample from the population always have the same distribution as the population? Let $Y_1,Y_2,...Y_n$ be a random sample of size $n$ from a population with distribution X.
From this information, can I also always conclude that $Y_1,Y_2,...Y_n$ will have distribution X? Or is this not the case? 
Is there a more formal mathematical way to show that this is necessarily true / false or depends on the situation?
 A: The answer to your question depends on what you mean when you refer to the "distribution" of the population and sample.  For either of these objects, the "distribution" can refer to an underlying probability distribution when the object is treated as a random variable, or it can refer to the actual empirical distribution of the values, when they are treated as fixed.
Clearly, it is not the case that the empirical distributions of the population and sample will match (unless of course, the sample is a full census).  However, there are certain useful equalities of "distribution" that occur between the population and sample under simple-random-sampling.  The relevant equalities depend on our approach to sampling theory.  If we have a finite population of $N$ values, there are two common methods of sampling analysis, and two corresponding ways we can talk about equality of the "distributions" of the population and sample.

Model-based analysis: Under this form of analysis we treat the population values as random variables from some underlying distribution, so that our population is $Y_1,..., Y_N \sim \text{IID }F$. The population values in this model are exchangeable random variables, so we can obtain a random sample of size $n$ by taking the first $n$ values in the sequence.  This means that we obtain the sample $Y_1,...,Y_n \sim \text{IID }F$, where the distribution stated here is unconditional on the population.  In this analysis the "distribution" of the population refers to the underlying distribution function $F$, and this is also the probability distribution of the sample (unconditional on the population values).
Design-based interpretation: Under this alternative form of analysis we treat the population values as fixed (unknown) quantities, so our population is a vector of scalar values $y_1,...,y_N$.  Since the population values are not treated as random variables, their "distribution" can only refer to the empirical distribution function $F_N$ defined by:
$$F_N(y) \equiv \frac{1}{N} \sum_{i=1}^N \mathbb{I}(y \leqslant y_i).$$
For a simple-random-sample, an individual sample value is formed by taking $y_{S(i)}$ where the $i$th sample index is $S(i) \sim \text{U} \{ 1,...,N \}$.  (For sampling with replacement these indices are independent; for sampling without replacement the indices cannot be repeated, and are therefore dependent.)  Conditional on the population values, we have $y_{S(i)}| y_1,...,y_N \sim F_N$, so the conditional probability distribution of a sample value, given the population values, is equal to the empirical distribution of the population values.

As you can see, in either case there is an equality of the "distributions" of the sample with the population, so long as these "distributions" are interpreted in the ways specified above.
A: As the sample size $n$ tends towards infinity, the sample distribution will approach the population distribution. When $n$ is small, it will often not look much like the population distribution at all.
For example, below are two random samples from the normal distribution, one with $n=10$ and one with $n=100$. 

