Whether you stay at UCLA or transfer somewhere else, you will need to take the lower division math courses:
Math 31A, 31B, 32A, 32B, 33A, 33B (these cover the same content in UCB Math 1A, 1B, 53, 54)
UCLA has additional requirements in other subjects besides math; see http://catalog.registrar.ucla.edu/uclacatalog2015-16-453.htm .
UCB requires a discrete math course (Math 55); UCLA Math 61 appears similar. See https://math.berkeley.edu/programs/undergraduate/major/pure .
You also need to complete at least some general education requirements, such as reading and composition and foreign language, or have a letter of reciprocity from UCLA after completing general education requirements there, for transfer purposes. But you need to do the UCLA general education requirements anyway if you stay there. See the following:
http://www.registrar.ucla.edu/ge/GE-LS15-16.pdf
http://ls-advise.berkeley.edu/requirement/summary.html
One of the most well known great mathematicians is currently on the faculty at UCLA.