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Theorem dmarea 23153
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 22045 . . . 4  |-  S. RR ( vol `  ( s
" { x }
) )  _d x  e.  _V
2 df-area 23152 . . . 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 5716 . . 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 2545 . 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 5338 . . . . . 6  |-  ( t  =  A  ->  (
t " { x } )  =  ( A " { x } ) )
65eleq1d 2536 . . . . 5  |-  ( t  =  A  ->  (
( t " {
x } )  e.  ( `' vol " RR ) 
<->  ( A " {
x } )  e.  ( `' vol " RR ) ) )
76ralbidv 2906 . . . 4  |-  ( t  =  A  ->  ( A. x  e.  RR  ( t " {
x } )  e.  ( `' vol " RR ) 
<-> 
A. x  e.  RR  ( A " { x } )  e.  ( `' vol " RR ) ) )
85fveq2d 5876 . . . . . 6  |-  ( t  =  A  ->  ( vol `  ( t " { x } ) )  =  ( vol `  ( A " {
x } ) ) )
98mpteq2dv 4540 . . . . 5  |-  ( t  =  A  ->  (
x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  =  ( x  e.  RR  |->  ( vol `  ( A
" { x }
) ) ) )
109eleq1d 2536 . . . 4  |-  ( t  =  A  ->  (
( x  e.  RR  |->  ( vol `  ( t
" { x }
) ) )  e.  L^1  <->  ( x  e.  RR  |->  ( vol `  ( A " { x }
) ) )  e.  L^1 ) )
117, 10anbi12d 710 . . 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 3266 . 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 9595 . . . . . 6  |-  RR  e.  _V
1413, 13xpex 6599 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
1514elpw2 4617 . . . 4  |-  ( A  e.  ~P ( RR 
X.  RR )  <->  A  C_  ( RR  X.  RR ) )
1615anbi1i 695 . . 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 977 . . 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 252 . 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 271 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821    C_ wss 3481   ~Pcpw 4016   {csn 4033    |-> cmpt 4511    X. cxp 5003   `'ccnv 5004   dom cdm 5005   "cima 5008   ` cfv 5594   RRcr 9503   volcvol 21743   L^1cibl 21894   S.citg 21895  areacarea 23151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-sum 13489  df-itg 21900  df-area 23152
This theorem is referenced by:  areambl  23154  areass  23155  areaf  23157  areacirc  30039  arearect  31112  areaquad  31113
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