Theorem dmarea 23748

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.

Step	Hyp	Ref	Expression
1		itgex 22605	. . . 4 |- S. RR ( vol ` ( s " { x } ) ) _d x e. _V
2		df-area 23747	. . . 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 )
3	1, 2	dmmpti 5725	. . 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 ) }
4	3	eleq2i 2507	. 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 5183	. . . . . 6 |- ( t = A -> ( t " { x } ) = ( A " { x } ) )
6	5	eleq1d 2498	. . . . 5 |- ( t = A -> ( ( t " { x } ) e. ( `' vol " RR ) <-> ( A " { x } ) e. ( `' vol " RR ) ) )
7	6	ralbidv 2871	. . . 4 |- ( t = A -> ( A. x e. RR ( t " { x } ) e. ( `' vol " RR ) <-> A. x e. RR ( A " { x } ) e. ( `' vol " RR ) ) )
8	5	fveq2d 5885	. . . . . 6 |- ( t = A -> ( vol ` ( t " { x } ) ) = ( vol ` ( A " { x } ) ) )
9	8	mpteq2dv 4513	. . . . 5 |- ( t = A -> ( x e. RR |-> ( vol ` ( t " { x } ) ) ) = ( x e. RR |-> ( vol ` ( A " { x } ) ) ) )
10	9	eleq1d 2498	. . . 4 |- ( t = A -> ( ( x e. RR |-> ( vol ` ( t " { x } ) ) ) e. L^1 <-> ( x e. RR |-> ( vol ` ( A " { x } ) ) ) e. L^1 ) )
11	7, 10	anbi12d 715	. . 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 ) ) )
12	11	elrab 3235	. 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 9629	. . . . . 6 |- RR e. _V
14	13, 13	xpex 6609	. . . . 5 |- ( RR X. RR ) e. _V
15	14	elpw2 4589	. . . 4 |- ( A e. ~P ( RR X. RR ) <-> A C_ ( RR X. RR ) )
16	15	anbi1i 699	. . 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 986	. . 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 ) ) )
18	16, 17	bitr4i 255	. 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 ) )
19	4, 12, 18	3bitri 274	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 ) )

Syntax hints:   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   A.wral 2782   {crab 2786   C_ wss 3442   ~Pcpw 3985   {csn 4002   |-> cmpt 4484   X. cxp 4852   `'ccnv 4853   dom cdm 4854   "cima 4857   ` cfv 5601   RRcr 9537   volcvol 22295   L^1cibl 22452   S.citg 22453  areacarea 23746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-sum 13731  df-itg 22458  df-area 23747

This theorem is referenced by:  areambl  23749  areass  23750  areaf  23752  areacirc  31744  arearect  35802  areaquad  35803