The recruiter, hopefully, was using your number as a filter to screen out positions he/she knows you will not be interested in. That would not be setting a cap on what you would like to make. If the market is set at that range, a recruiter, who is often paid a fee based on the first year's compensation, would be foolish to offer a candidate for less than the minimum of the range.
Also, most companies are going to offer the lower bound, but not less. They deemed that to be the position's market-determined range, for the skills and experience wanted. Most companies might have some sort of extraordinary process to bring someone in above range, but if you're worth less than the range, they would deem you not qualified for that position.
Also, if someone comes to work and they do equal or better work than co-workers who are being paid 30% to 60% more, they know they are going to lose that worker, or have a worker who feels very taken advantage of. Dealing with an opening and then actually filling that opening is a very expensive process, so they don't want to lose money, long-term, by shorting someone of compensation, short-term.
Also, if they get a reputation as a company that drastically under-compensates employees when the opportunity arises, they are not going to be able to bring top talent on board for other positions.
If companies are able to bring someone qualified on board at the minimum, and they're happy about it, that's a huge win for the company. If a recruiter is able to offer someone to a company who is qualified that they know will be happy starting at the bottom of a position's expected range, then that's a very hassle-free placement and commission earned, and they look like they are awesome recruiters in the eyes of the company, which means more business or even, possibly, a more exclusive or favored relationship with the hiring manager.
You should be fine. If the recruiter was planning on bringing you in below the lowest pay level for a position, that recruiter would not have disclosed that the "floor" for that position was higher than the number you targeted. Can't hurt to confirm that your expectation is, for a position that is a higher paying job than the minimum your laid out, that the recruiter will not try to offer you as a "low-ball" for the company.
