I think you'd be better off with a gtx 1060, save a buck.
Find some benchmarks on the two and compare.
Here is my current build:
-AMD FX 6100 Six-core 1.5 Ghz CPU
-ASUS M5A97 R2.0 Motherboard
-AMD Radeon R9 20 Series 3Gb GPU
-8Gb of DDR3 RAM
-Corsair VS650 Power Supply
-500mb Boot SSD
-1Tb Corsair Hard Drive
My question is this: Can I upgrade to an AMD Radeon rx 580 GPU without upgrading any other components, and if I must, which compononents would those be. I saw a benchmark for the rx 580 GPU and it could run BF1 FullHD 1080p on Ultra graphics at an average of 90fps, my goal is to be able to run it on this same preset with minimal drops under 60fps.

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic