"A/the" before library and gallery

[Rachel Adams]

If I know that there is only one gallery and only one library in my city and if I mention either of them for the first time should I use the definite or the indefinite article?

For example, 1. "I went to the art gallery yesterday." 2. "I went to the library yesterday." I think I should use "the".

 

[Rachel Adams]

'The' is fine.

But "a" is wrong, isn't it?