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Theorem areass 20307
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 20305 . 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 970 1  |-  ( S  e.  dom area  ->  S  C_  ( RR  X.  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1701   A.wral 2577    C_ wss 3186   {csn 3674    e. cmpt 4114    X. cxp 4724   `'ccnv 4725   dom cdm 4726   "cima 4729   ` cfv 5292   RRcr 8781   volcvol 18876   L ^1cibl 19025  areacarea 20303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-fv 5300  df-sum 12206  df-itg 19032  df-area 20304
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