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Their comments come at an eventful crossroads for District-congressional relations, as lawmakers on Capitol Hill tried to avoid a government shutdown that could negatively impact day-to-day activity in the city.

Mr. Issa said he is committed to forging a bill that gives the District more autonomy over its own spending, allowing it to avoid the tumult tied to wrangling over federal appropriations. D.C. officials rejected an initial proposal by Mr. Issa last month, citing a provision that makes permanent a ban on the city from using public dollars to fund abortions.