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Theorem dmarea 20268
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 19141 . . . 4  |-  S. RR ( vol `  ( s
" { x }
) )  _d x  e.  _V
2 df-area 20267 . . . 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 5389 . . 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 2360 . 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 5023 . . . . . 6  |-  ( t  =  A  ->  (
t " { x } )  =  ( A " { x } ) )
65eleq1d 2362 . . . . 5  |-  ( t  =  A  ->  (
( t " {
x } )  e.  ( `' vol " RR ) 
<->  ( A " {
x } )  e.  ( `' vol " RR ) ) )
76ralbidv 2576 . . . 4  |-  ( t  =  A  ->  ( A. x  e.  RR  ( t " {
x } )  e.  ( `' vol " RR ) 
<-> 
A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR ) ) )
85fveq2d 5545 . . . . . 6  |-  ( t  =  A  ->  ( vol `  ( t " { x } ) )  =  ( vol `  ( A " {
x } ) ) )
98mpteq2dv 4123 . . . . 5  |-  ( t  =  A  ->  (
x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  =  ( x  e.  RR  |->  ( vol `  ( A
" { x }
) ) ) )
109eleq1d 2362 . . . 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 2936 . 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 8844 . . . . . 6  |-  RR  e.  _V
1413, 13xpex 4817 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
1514elpw2 4191 . . . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   ~Pcpw 3638   {csn 3653    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   dom cdm 4705   "cima 4708   ` cfv 5271   RRcr 8752   volcvol 18839   L ^1cibl 18988   S.citg 18989  areacarea 20266
This theorem is referenced by:  areambl  20269  areass  20270  areaf  20272  areacirc  25034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-sum 12175  df-itg 18995  df-area 20267
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