How do I select an element only when inside another element?

I have a question regarding CSS selectors. How do I select a <div> with a specific class name only when its inside a <ul> with a class name saft? This CSS class is used elsewhere and I don't want to change the styling everywhere.

<div id="floater">
    <ul class="saft">
        <li><div class="textSlide"></li>
    </ul>
</div>

Simply use the CSS descendant selector (a space) between the parent element and the descendant element:

ul.saft div.textSlide {
    /* CSS rules */
}

JS Fiddle demo.

In this case the rules applied to the div of class textSlide will only apply if its ancestor is a ul of the class saft. You could, instead, use the immediate child combinator, the > but then you'd have to specify the div in relation to each parent/ancestor up to the one whose class is required, which gives potentially difficult to maintain CSS (not in this case, but it can, over time, become problematic).

Just do:

ul.saft .textSlide {

This achieves what you need