The Committee released a statement claiming that the agency that would implement Obama’s executive action — U.S. Citizenship and Immigration Services — is entirely self-funded through fees it collects, and could continue its operations even if the government isn’t funded through the appropriations process. “The appropriations process cannot be used to ‘defund’ the agency,” the statement said.

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Game over? Well, no.

There are still ways for Republicans to proceed. Ted Cruz, for one, is not calling for a direct “defunding” of the agency that would implement Obama’s “executive amnesty.” Rather, he argues that Republicans should attach “riders” to the funding of other government functions — “critical” ones — that would also “strip the authority from the president to grant amnesty.” If Obama vetoed those measures, he would then take the blame for shutting down those functions of government. Thus, Republicans are the ones with the leverage. Eventually he’d have to buckle and give up his amnesty. Or so Cruz believes, anyway.

Alternatively, Republicans could attach riders to appropriations bills that also restrict funding from being used to carry out Obama’s executive action. Even if those funds are generated by fees, Congress can restrict their use for particular measures.

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I asked Jennifer Hing, a spokesperson for the House Appropriations Committee, to comment on the rider strategy. She emailed:

“Chairman Rogers is adamantly opposed to the President’s purported executive action on immigration, and wants Congress to fight it tooth and nail through legislative means. However, if a rider on the executive order was attached to an Omnibus Appropriations bill, the President would veto it, thus shutting down the federal government. In that scenario, the Citizenship and Immigration Service – the very agency responsible for implementing the President’s order – could continue to operate. Therefore, the President’s order would likely be untouched, and potentially millions of undocumented people could get immigration status, while the rest of the government is shuttered. That strategy makes no sense.”

And so, yes, Republicans could pursue the rider strategy. But Obama would veto it, and we would get…another government shutdown fight. Meanwhile, Obama’s amnesty would continue.

Of course, conservatives will demand that GOP leaders pursue this strategy, because they appear to believe that if Republicans hang in there long enough, the shuttering of important government functions will ultimately put pressure on Obama to relent. It’s very possible that the politics of this fight could get very dicey for Democrats. You could see a situation in which a handful of Senate Democrats (who support legislative reform but are uncomfortable with the substance or politics of unilateral action) initially defect and side with Republicans against Obama’s move. These Democrats might feel pressure to vote for government funding measures that contain anti-amnesty riders.

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But even if that happened, Obama could continue to hold the line with the veto. Which is to say that this would all but certainly turn into a good old-fashioned government shutdown fight. As Jonathan Bernstein has explained, such fights structurally favor the president over Congress. And in this case, Republicans might get maneuvered into a position where the public sees them creating havoc solely because they oppose Obama’s efforts to shield millions of undocumented immigrants from deportation. Those who hope to broaden the party’s appeal heading into 2016 probably don’t want that to happen.