MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  areass Structured version   Unicode version

Theorem areass 23750
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 23748 . 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 1020 1  |-  ( S  e.  dom area  ->  S  C_  ( RR  X.  RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   A.wral 2782    C_ wss 3442   {csn 4002    |-> cmpt 4484    X. cxp 4852   `'ccnv 4853   dom cdm 4854   "cima 4857   ` cfv 5601   RRcr 9537   volcvol 22295   L^1cibl 22452  areacarea 23746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-sum 13731  df-itg 22458  df-area 23747
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator