Which variables matter most for prediction of another variable?
Let's use the following notation : $(x_1, ...x_9)$ are the first 9 variables out of which you try to predict $y$, your tenth variable.

I will try to address what looks like a PCA misunderstanding, then give you some ways to predict which variables $ x_i $ matter most to predict $y$

Principal Component Analysis 

The returned features you get from the PCA are not the original one! 

The PCA transforms $(x_1, ..., x_9)$ into $(x'_1, ..., x'_9)$ where each $x'_i$ is a particular combination of the $(x_1, ..., x_9)$ such that for any $i, i'$, $x_i$ and $x_i'$ are linearly uncorrelated (ie orthogonal). Moreover, it is done such that each $x_i$ explains a certain part of the training dataset variance. In fact, most of the implementation sort the $x'_1, ..., x'_9$ such that the first one is the one explaining the variance the most, then the second, ...

Thus you end up having your first variable $x'_1$ explaining a lot your prediction $y$. But this $x'_i$ is a particular combinaison of $(x_1, ..., x_9)$.

Getting the feature importance

As to assert which variable $x_i$ explains $y$ the best, one might use different ways. Just to mention that the feature importance is not absolute ; it relies on the technique (or estimator) you use to address this question. Here is a non exhaustive list of possibilities :


The linear correlation is a good start
With the RandomForestClassifier (or RandomForestRegressor depending on $y$) of sklearn.ensemble, you can use feature_importances_ method to get which one is used the most at tree nodes. Note : it works with any decision tree estimator.
With standard linear or logistic regression (same with Lasso, Ridge, ...), you can check which variable has the higher coefficient (do not forget to normalize your input variables $(x_1, ..., x_9)$ ) 


I don't know how it is done using neural networks but would be glad if anyone has an hint.