Where does the phrase "That's a wrap" come from?

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The O.E.D.'s first citation is from a 1974 cinematographic novel by Michael Ayrton: "Other cars are heard starting up out of shot and the lights on the pergola go off so I assume it's a wrap and the crew is listening to the director saying something consequential and busy about tomorrow's call."

However, assiduous research turns up this 1957 entry in Charlton Heston's journal, quoted in the 1998 edition of "This Is Orson Welles," by Welles and Peter Bogdanovich: "We rehearsed all day . . . the studio brass gathering in the shadows in anxious little knots. By the time we began filming at 5:45, I knew they'd written off the whole day. At 7:40, Orson said: 'O.K., print. That's a wrap on this set.' "

Thus, it appears that the switch from wrap it up to that's a wrap took place in the 50's. That seems to make its use in "The Aviator" about a party in 1930 an anachronism (from the Greek ana, "back," and chronos, "time")."
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William Safire; "It's a Wrap," The New York Times, Feb. 27, 2005

The 1974 citation quoted is still the earliest in the OED.

A quick look in Google Books turned up:

Where you have five or six union guys sitting around who do nothing but jack up the cost. There's always one guy who invariably falls sound asleep before the second take and doesn't wake up until it's a wrap. Marketing/Communications. Snippet view 1971(?)

as well as this similar, earlier version:

"...I think it's a wrap-up. Now everybody take a breather, will you? Four-thirty run-through, please. Will you all kindly get back here promptly?” People drifted off the set, the musicians following with slothlike languor. Monte Sohn; The Flesh and Mary Duncan (1948) Snippet view

It could be that "It's a wrap-up" (perhaps from "Let's wrap it up!") was the first version. And it may have first been used in the theater.

The noun comes from the verb "to wrap"

OED

2.a. To cover or envelop (an object) by winding or folding something round or about it; to surround with or enwrap in a covering, wrapper, or the like, esp. so as to protect from injury, damage, loss, etc.

In the transferred sense, to wrap something is the final action in dealing with it.

This gave rise to:

7. to wrap up (figurative).

a. transitive. To put an end to, bring to completion; also, to defeat; to wrap it up, to stop doing something. slang.

1922 T. E. Lawrence Seven Pillars (Oxford text) xxxviii. 90 The British were wrapping up the Arabs on all sides—at Aden, at Gaza, at Bagdad.

1937 Amer. Legion Monthly May 9/1 Only one shot to finish before midnight and we'd wrap it up in thirteen days.

Hence "That is a wrap" - "That is finished".