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There was a video camera mounted and extended from the side of the landing module especially for this purpose. NASA anticipated that the moment that Armstrong stepped onto the Moon would be thought as being particularly significant and something everyone would want to see.
As Armstrong started down the ladder he remotely deployed it, swinging it out from its storage position in the side of the module. Note how the left hand side of the video image is obscured by both the side of the Lunar Module and the arm of the storage compartment that lowered it. (The horizontal black bar across the middle is interference flickering across the TV screen it was filmed off.)
The Lunar Module did! While still on the steps, Armstrong deployed the Modularized Equipment Stowage Assembly from the side of the lunar module.This housed, amongst other things, the TV camera. This meant that upward of 600 million people on earth could take part in this historic moment.