Etymology
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federal (adj.)

1640s, as a theological term (in reference to "covenants" between God and man), from French fédéral, an adjective formed from Latin foedus (genitive foederis) "covenant, league, treaty, alliance" (from PIE *bhoid-es-, suffixed form of root *bheidh- "to trust, confide, persuade").

Secular meaning "pertaining to a covenant or treaty" (1650s) led to political sense of "formed by agreement among independent states" (1707), from use of the word in federal union "union based on a treaty" (popularized during formation of U.S.A. 1776-1787) and like phrases. Also from this period in U.S. history comes the sense "favoring the central government" (1788) and the especial use of the word (as opposed to confederate) to mean a state in which the federal authority is independent of the component parts within its legitimate sphere of action. Used from 1861 in reference to the Northern forces in the American Civil War.

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fed (n.)
1788, short for Federalist; as colloquial for "official of the federal government," from 1916; especially, since 1930s, of FBI agents.
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federalism (n.)
1788, "doctrine of federal union in government," American English, from French fédéralisme, from fédéral (see federal). Also, from about the same time and place, "doctrines of the Federalist Party in American politics."
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federalist (n.)
1787, American English, "member or supporter of the Federal party in U.S. politics" (originally of supporters of the Philadelphia constitution), from federal + -ist. General sense of "one who supports federal union" is from 1792. The party expired c. 1824. As an adjective by 1801.
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*bheidh- 
Proto-Indo-European root meaning "to trust, confide, persuade."

It forms all or part of: abide; abode; affiance; affidavit; auto-da-fe; bide; bona fide; confederate; confidant; confide; confidence; confident; defiance; defy; diffidence; diffident; faith; fealty; federal; federate; federation; fiancee; fideism; fidelity; fiducial; fiduciary; infidel; infidelity; nullifidian; perfidy; solifidian.

It is the hypothetical source of/evidence for its existence is provided by: Greek pistis "faith, confidence, honesty;" Latin fides "trust, faith, confidence, reliance, credence, belief;" Albanian be "oath," bindem "to be convinced, believe;" Old Church Slavonic beda "distress, necessity," bediti "to force, persuade;" Old English biddan "to ask, beg, pray," German bitten "to ask."
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FCC 
U.S. Federal Communications Commission, formed 1934 from the former Federal Radio Commission.
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Freddie Mac 
by 1992, vaguely from Federal Home Loan Mortgage Corporation.
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FBI 
U.S. Federal Bureau of Investigation, formed 1935 from the former United States Bureau of Investigation (1908).
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Fannie Mae (n.)
1948, from FNMA, acronym of "Federal National Mortgage Association," established 1938.
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Bundestag (n.)
German federal council, 1879, from German Bundestag, from genitive of Bund "league, confederacy, association" (related to English band (n.2) and bind (v.)) + tag, literally "day;" as a verb, tagen, "to sit in conference" (see day; also compare adjourn). Hence also Bundesrat "federal council of the German empire" (1872), from rat, rath "council" (see rathskeller).
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