Theorem dmarea 24684

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 23537	. . . 4 |- S. RR ( vol ` ( s " { x } ) ) _d x e. _V
2		df-area 24683	. . . 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 6023	. . 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 2693	. 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 5461	. . . . . 6 |- ( t = A -> ( t " { x } ) = ( A " { x } ) )
6	5	eleq1d 2686	. . . . 5 |- ( t = A -> ( ( t " { x } ) e. ( `' vol " RR ) <-> ( A " { x } ) e. ( `' vol " RR ) ) )
7	6	ralbidv 2986	. . . 4 |- ( t = A -> ( A. x e. RR ( t " { x } ) e. ( `' vol " RR ) <-> A. x e. RR ( A " { x } ) e. ( `' vol " RR ) ) )
8	5	fveq2d 6195	. . . . . 6 |- ( t = A -> ( vol ` ( t " { x } ) ) = ( vol ` ( A " { x } ) ) )
9	8	mpteq2dv 4745	. . . . 5 |- ( t = A -> ( x e. RR |-> ( vol ` ( t " { x } ) ) ) = ( x e. RR |-> ( vol ` ( A " { x } ) ) ) )
10	9	eleq1d 2686	. . . 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 747	. . 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 3363	. 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 10027	. . . . . 6 |- RR e. _V
14	13, 13	xpex 6962	. . . . 5 |- ( RR X. RR ) e. _V
15	14	elpw2 4828	. . . 4 |- ( A e. ~P ( RR X. RR ) <-> A C_ ( RR X. RR ) )
16	15	anbi1i 731	. . 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 1042	. . 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 267	. 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 286	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   ` cfv 5888   RRcr 9935   volcvol 23232   L^1cibl 23386   S.citg 23387  areacarea 24682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-sum 14417  df-itg 23392  df-area 24683

This theorem is referenced by:  areambl  24685  areass  24686  areaf  24688  areacirc  33505  arearect  37801  areaquad  37802