If your mechanic hesitated, it's probably because they were talking to you off the top of their head without researching it. It's not like all of us can remember instantly everything we ever have seen or knew.
You could always just ask them to verify the compatibility and show you. In the 2010-2011 compatibility archive they show the RD-5700 compatible with ST-5600 shifters if those are what you have.
https://productinfo.shimano.com/down...ibility_en.pdf
5701 came out in a following year. And the 5600 stuff disappeared in the meantime from documentation as they were no longer made. However you can infer that the 5701 will work with your 5600 stuff. Your only concern might be that the 5701 DR specs for front max difference and the largest rear cog are within what your bike is. However it likely is okay, unless you've put a cassette on it with a really big tooth count low cog.