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Theorem areass 20798
Description: A measurable region is a subset of  RR  X.  RR. (Contributed by Mario Carneiro, 21-Jun-2015.)
Assertion
Ref Expression
areass  |-  ( S  e.  dom area  ->  S  C_  ( RR  X.  RR ) )

Proof of Theorem areass
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dmarea 20796 . 2  |-  ( S  e.  dom area  <->  ( S  C_  ( RR  X.  RR )  /\  A. x  e.  RR  ( S " { x } )  e.  ( `' vol " RR )  /\  (
x  e.  RR  |->  ( vol `  ( S
" { x }
) ) )  e.  L ^1 ) )
21simp1bi 972 1  |-  ( S  e.  dom area  ->  S  C_  ( RR  X.  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   A.wral 2705    C_ wss 3320   {csn 3814    e. cmpt 4266    X. cxp 4876   `'ccnv 4877   dom cdm 4878   "cima 4881   ` cfv 5454   RRcr 8989   volcvol 19360   L ^1cibl 19509  areacarea 20794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-sum 12480  df-itg 19516  df-area 20795
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