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Theorem dmarea 20252
Description: The domain of the area function is the set of finitely measurable subsets of  RR  X.  RR. (Contributed by Mario Carneiro, 21-Jun-2015.)
Assertion
Ref Expression
dmarea  |-  ( A  e.  dom area  <->  ( A  C_  ( RR  X.  RR )  /\  A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR )  /\  (
x  e.  RR  |->  ( vol `  ( A
" { x }
) ) )  e.  L ^1 ) )
Distinct variable group:    x, A

Proof of Theorem dmarea
Dummy variables  t 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itgex 19125 . . . 4  |-  S. RR ( vol `  ( s
" { x }
) )  _d x  e.  _V
2 df-area 20251 . . . 4  |- area  =  ( s  e.  { t  e.  ~P ( RR 
X.  RR )  |  ( A. x  e.  RR  ( t " { x } )  e.  ( `' vol " RR )  /\  (
x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  e.  L ^1 ) } 
|->  S. RR ( vol `  ( s " {
x } ) )  _d x )
31, 2dmmpti 5373 . . 3  |-  dom area  =  {
t  e.  ~P ( RR  X.  RR )  |  ( A. x  e.  RR  ( t " { x } )  e.  ( `' vol " RR )  /\  (
x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  e.  L ^1 ) }
43eleq2i 2347 . 2  |-  ( A  e.  dom area  <->  A  e.  { t  e.  ~P ( RR 
X.  RR )  |  ( A. x  e.  RR  ( t " { x } )  e.  ( `' vol " RR )  /\  (
x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  e.  L ^1 ) } )
5 imaeq1 5007 . . . . . 6  |-  ( t  =  A  ->  (
t " { x } )  =  ( A " { x } ) )
65eleq1d 2349 . . . . 5  |-  ( t  =  A  ->  (
( t " {
x } )  e.  ( `' vol " RR ) 
<->  ( A " {
x } )  e.  ( `' vol " RR ) ) )
76ralbidv 2563 . . . 4  |-  ( t  =  A  ->  ( A. x  e.  RR  ( t " {
x } )  e.  ( `' vol " RR ) 
<-> 
A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR ) ) )
85fveq2d 5529 . . . . . 6  |-  ( t  =  A  ->  ( vol `  ( t " { x } ) )  =  ( vol `  ( A " {
x } ) ) )
98mpteq2dv 4107 . . . . 5  |-  ( t  =  A  ->  (
x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  =  ( x  e.  RR  |->  ( vol `  ( A
" { x }
) ) ) )
109eleq1d 2349 . . . 4  |-  ( t  =  A  ->  (
( x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  e.  L ^1  <->  ( x  e.  RR  |->  ( vol `  ( A " { x }
) ) )  e.  L ^1 ) )
117, 10anbi12d 691 . . 3  |-  ( t  =  A  ->  (
( A. x  e.  RR  ( t " { x } )  e.  ( `' vol " RR )  /\  (
x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  e.  L ^1 )  <->  ( A. x  e.  RR  ( A " { x }
)  e.  ( `' vol " RR )  /\  ( x  e.  RR  |->  ( vol `  ( A " { x }
) ) )  e.  L ^1 ) ) )
1211elrab 2923 . 2  |-  ( A  e.  { t  e. 
~P ( RR  X.  RR )  |  ( A. x  e.  RR  ( t " {
x } )  e.  ( `' vol " RR )  /\  ( x  e.  RR  |->  ( vol `  (
t " { x } ) ) )  e.  L ^1 ) }  <->  ( A  e. 
~P ( RR  X.  RR )  /\  ( A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR )  /\  ( x  e.  RR  |->  ( vol `  ( A " { x }
) ) )  e.  L ^1 ) ) )
13 reex 8828 . . . . . 6  |-  RR  e.  _V
1413, 13xpex 4801 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
1514elpw2 4175 . . . 4  |-  ( A  e.  ~P ( RR 
X.  RR )  <->  A  C_  ( RR  X.  RR ) )
1615anbi1i 676 . . 3  |-  ( ( A  e.  ~P ( RR  X.  RR )  /\  ( A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR )  /\  ( x  e.  RR  |->  ( vol `  ( A " { x }
) ) )  e.  L ^1 ) )  <-> 
( A  C_  ( RR  X.  RR )  /\  ( A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR )  /\  ( x  e.  RR  |->  ( vol `  ( A " { x }
) ) )  e.  L ^1 ) ) )
17 3anass 938 . . 3  |-  ( ( A  C_  ( RR  X.  RR )  /\  A. x  e.  RR  ( A " { x }
)  e.  ( `' vol " RR )  /\  ( x  e.  RR  |->  ( vol `  ( A " { x }
) ) )  e.  L ^1 )  <->  ( A  C_  ( RR  X.  RR )  /\  ( A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR )  /\  (
x  e.  RR  |->  ( vol `  ( A
" { x }
) ) )  e.  L ^1 ) ) )
1816, 17bitr4i 243 . 2  |-  ( ( A  e.  ~P ( RR  X.  RR )  /\  ( A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR )  /\  ( x  e.  RR  |->  ( vol `  ( A " { x }
) ) )  e.  L ^1 ) )  <-> 
( A  C_  ( RR  X.  RR )  /\  A. x  e.  RR  ( A " { x }
)  e.  ( `' vol " RR )  /\  ( x  e.  RR  |->  ( vol `  ( A " { x }
) ) )  e.  L ^1 ) )
194, 12, 183bitri 262 1  |-  ( A  e.  dom area  <->  ( A  C_  ( RR  X.  RR )  /\  A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR )  /\  (
x  e.  RR  |->  ( vol `  ( A
" { x }
) ) )  e.  L ^1 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   ~Pcpw 3625   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692   ` cfv 5255   RRcr 8736   volcvol 18823   L ^1cibl 18972   S.citg 18973  areacarea 20250
This theorem is referenced by:  areambl  20253  areass  20254  areaf  20256  areacirc  24931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-sum 12159  df-itg 18979  df-area 20251
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