If i bring my router from my old apartment to a new one, am I still connected to my old internet service?

[Matt The person]

No. What you're seeing is that you're connected to the same network since it's the same router. The purpose of a router is to deliver internet to multiple devices using one connection. You can still have the same network name (SSID), but have a different Internet Service Provider. (See diagram Below)

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ISP --|> Modem --> Router --> Client (The device connected to the internet, ie. iPhone, Laptop, etc)
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Basically what I'm trying to get at with the diagram is that even when your Internet Service Provider (ISP) changes, your network name on your router still the same. Unless of course you change it yourself.