Air carrier Captain reported being in close proximity to another aircraft during taxi at DFW and having to apply the brakes aggressively. Taxi was continued.

Narrative

Taxiing out into the sunrise; ATC cleared us from Gate XX over the A bridge via G; G10; F; A. As we passed the B taxiway bridge I noted an approaching Aircraft Y; and assumed they would give way due to our aforementioned thru taxi clearance. As I made the turn onto alpha my attention was directed down to the lead line and as I was completing the turn I looked up and noted Aircraft Y had turned from B to G and was still moving. Due to our close proximity I had to apply the brakes more aggressively than usual. We held position for a moment then ATC told Aircraft Y to give way to us. As we resumed our taxi I instructed the FO; Person A; to call the FA's and determine if everyone was okay. He made the call and I was informed that FA Person B had fallen; but he was claiming to feel fine. Once we were in a position to stop; I made the same call and spoke to Person A myself. He made the same claim that he felt ok and wanted to continue. With this information we elected to continue the flight.Cause: ATC did not warn us of any potential conflicts; and we assumed the other craft would give way. The morning sun was driving into my window on that southern facing taxi making it difficult to see for myself.