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Theorem dmarea 23932
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 22777 . . . 4  |-  S. RR ( vol `  ( s
" { x }
) )  _d x  e.  _V
2 df-area 23931 . . . 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 5729 . . 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 2532 . 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 5182 . . . . . 6  |-  ( t  =  A  ->  (
t " { x } )  =  ( A " { x } ) )
65eleq1d 2524 . . . . 5  |-  ( t  =  A  ->  (
( t " {
x } )  e.  ( `' vol " RR ) 
<->  ( A " {
x } )  e.  ( `' vol " RR ) ) )
76ralbidv 2839 . . . 4  |-  ( t  =  A  ->  ( A. x  e.  RR  ( t " {
x } )  e.  ( `' vol " RR ) 
<-> 
A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR ) ) )
85fveq2d 5892 . . . . . 6  |-  ( t  =  A  ->  ( vol `  ( t " { x } ) )  =  ( vol `  ( A " {
x } ) ) )
98mpteq2dv 4504 . . . . 5  |-  ( t  =  A  ->  (
x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  =  ( x  e.  RR  |->  ( vol `  ( A
" { x }
) ) ) )
109eleq1d 2524 . . . 4  |-  ( t  =  A  ->  (
( x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  e.  L^1  <->  ( x  e.  RR  |->  ( vol `  ( A " { x }
) ) )  e.  L^1 ) )
117, 10anbi12d 722 . . 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 3208 . 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 9656 . . . . . 6  |-  RR  e.  _V
1413, 13xpex 6622 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
1514elpw2 4581 . . . 4  |-  ( A  e.  ~P ( RR 
X.  RR )  <->  A  C_  ( RR  X.  RR ) )
1615anbi1i 706 . . 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 995 . . 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 260 . 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 279 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 setvar class
Syntax hints:    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   A.wral 2749   {crab 2753    C_ wss 3416   ~Pcpw 3963   {csn 3980    |-> cmpt 4475    X. cxp 4851   `'ccnv 4852   dom cdm 4853   "cima 4856   ` cfv 5601   RRcr 9564   volcvol 22464   L^1cibl 22624   S.citg 22625  areacarea 23930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-sum 13802  df-itg 22630  df-area 23931
This theorem is referenced by:  areambl  23933  areass  23934  areaf  23936  areacirc  32082  arearect  36145  areaquad  36146
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