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Theorem dmarea 20797
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 19663 . . . 4  |-  S. RR ( vol `  ( s
" { x }
) )  _d x  e.  _V
2 df-area 20796 . . . 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 5575 . . 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 2501 . 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 5199 . . . . . 6  |-  ( t  =  A  ->  (
t " { x } )  =  ( A " { x } ) )
65eleq1d 2503 . . . . 5  |-  ( t  =  A  ->  (
( t " {
x } )  e.  ( `' vol " RR ) 
<->  ( A " {
x } )  e.  ( `' vol " RR ) ) )
76ralbidv 2726 . . . 4  |-  ( t  =  A  ->  ( A. x  e.  RR  ( t " {
x } )  e.  ( `' vol " RR ) 
<-> 
A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR ) ) )
85fveq2d 5733 . . . . . 6  |-  ( t  =  A  ->  ( vol `  ( t " { x } ) )  =  ( vol `  ( A " {
x } ) ) )
98mpteq2dv 4297 . . . . 5  |-  ( t  =  A  ->  (
x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  =  ( x  e.  RR  |->  ( vol `  ( A
" { x }
) ) ) )
109eleq1d 2503 . . . 4  |-  ( t  =  A  ->  (
( x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  e.  L ^1  <->  ( x  e.  RR  |->  ( vol `  ( A " { x }
) ) )  e.  L ^1 ) )
117, 10anbi12d 693 . . 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 3093 . 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 9082 . . . . . 6  |-  RR  e.  _V
1413, 13xpex 4991 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
1514elpw2 4365 . . . 4  |-  ( A  e.  ~P ( RR 
X.  RR )  <->  A  C_  ( RR  X.  RR ) )
1615anbi1i 678 . . 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 941 . . 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 245 . 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 264 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   {crab 2710    C_ wss 3321   ~Pcpw 3800   {csn 3815    e. cmpt 4267    X. cxp 4877   `'ccnv 4878   dom cdm 4879   "cima 4882   ` cfv 5455   RRcr 8990   volcvol 19361   L ^1cibl 19510   S.citg 19511  areacarea 20795
This theorem is referenced by:  areambl  20798  areass  20799  areaf  20801  areacirc  26298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463  df-sum 12481  df-itg 19517  df-area 20796
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