androidx.legacy:legacy-support-v4 is Androidx artifacts of com.android.support:support-v4
com.android.support:support-v13 -> androidx.legacy:legacy-support-v13
com.android.support:support-v4 -> androidx.legacy:legacy-support-v4
You can find info about the library mapping here
The Support Library is a static library that you can add to your Android application in order to use APIs that are either not available for older platform versions or utility APIs that aren’t a part of the framework APIs. Compatible on devices running API 14 or later.
Uses for the Support Libraries
There are a few distinct uses for the support libraries. Backward compatibility classes for earlier versions of the platform is just one of them.
- Backward Compatibility for newer APIs – A large amount of the support libraries provide backward compatibility for newer framework classes and methods. For example, the Fragment support class provides support for fragments on devices running versions earlier than Android 3.0 (API level 11).
- Convenience and Helper Classes – The support libraries provides a number of helper classes, particularly for user interface development. For example, the RecyclerView class provides a user interface widget for displaying and managing very long lists, useable on versions of Android from API level 7 and up.
- Debugging and Utilities – There are a number of features that provide utility beyond code you incorporate into your app, including the support-annotations library for improved code lint checks on method inputs and Multidex support for configuring and distributing apps with over 65,536 methods.
See official documents here support-library