Metamath Proof Explorer

Theorem dmarea

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)

Ref Expression
Assertion 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 ) )

Proof

Step Hyp Ref Expression
1 itgex 
 |-  S. RR ( vol ` ( s " { x } ) ) _d x e. _V
2 df-area 
 |-  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 )
3 1 2 dmmpti 
 |-  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 ) }
4 3 eleq2i 
 |-  ( 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 
 |-  ( t = A -> ( t " { x } ) = ( A " { x } ) )
6 5 eleq1d 
 |-  ( t = A -> ( ( t " { x } ) e. ( `' vol " RR ) <-> ( A " { x } ) e. ( `' vol " RR ) ) )
7 6 ralbidv 
 |-  ( t = A -> ( A. x e. RR ( t " { x } ) e. ( `' vol " RR ) <-> A. x e. RR ( A " { x } ) e. ( `' vol " RR ) ) )
8 5 fveq2d 
 |-  ( t = A -> ( vol ` ( t " { x } ) ) = ( vol ` ( A " { x } ) ) )
9 8 mpteq2dv 
 |-  ( t = A -> ( x e. RR |-> ( vol ` ( t " { x } ) ) ) = ( x e. RR |-> ( vol ` ( A " { x } ) ) ) )
10 9 eleq1d 
 |-  ( t = A -> ( ( x e. RR |-> ( vol ` ( t " { x } ) ) ) e. L^1 <-> ( x e. RR |-> ( vol ` ( A " { x } ) ) ) e. L^1 ) )
11 7 10 anbi12d 
 |-  ( 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 ) ) )
12 11 elrab 
 |-  ( 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 
 |-  RR e. _V
14 13 13 xpex 
 |-  ( RR X. RR ) e. _V
15 14 elpw2 
 |-  ( A e. ~P ( RR X. RR ) <-> A C_ ( RR X. RR ) )
16 15 anbi1i 
 |-  ( ( 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 
 |-  ( ( 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 ) ) )
18 16 17 bitr4i 
 |-  ( ( 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 ) )
19 4 12 18 3bitri 
 |-  ( 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 ) )