Theorem dmarea 23932

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 22777	. . . 4 |- S. RR ( vol ` ( s " { x } ) ) _d x e. _V
2		df-area 23931	. . . 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 5729	. . 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 2532	. 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 5182	. . . . . 6 |- ( t = A -> ( t " { x } ) = ( A " { x } ) )
6	5	eleq1d 2524	. . . . 5 |- ( t = A -> ( ( t " { x } ) e. ( `' vol " RR ) <-> ( A " { x } ) e. ( `' vol " RR ) ) )
7	6	ralbidv 2839	. . . 4 |- ( t = A -> ( A. x e. RR ( t " { x } ) e. ( `' vol " RR ) <-> A. x e. RR ( A " { x } ) e. ( `' vol " RR ) ) )
8	5	fveq2d 5892	. . . . . 6 |- ( t = A -> ( vol ` ( t " { x } ) ) = ( vol ` ( A " { x } ) ) )
9	8	mpteq2dv 4504	. . . . 5 |- ( t = A -> ( x e. RR |-> ( vol ` ( t " { x } ) ) ) = ( x e. RR |-> ( vol ` ( A " { x } ) ) ) )
10	9	eleq1d 2524	. . . 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 722	. . 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 3208	. 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 9656	. . . . . 6 |- RR e. _V
14	13, 13	xpex 6622	. . . . 5 |- ( RR X. RR ) e. _V
15	14	elpw2 4581	. . . 4 |- ( A e. ~P ( RR X. RR ) <-> A C_ ( RR X. RR ) )
16	15	anbi1i 706	. . 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 995	. . 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 260	. 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 279	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 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   A.wral 2749   {crab 2753   C_ wss 3416   ~Pcpw 3963   {csn 3980   |-> cmpt 4475   X. cxp 4851   `'ccnv 4852   dom cdm 4853   "cima 4856   ` cfv 5601   RRcr 9564   volcvol 22464   L^1cibl 22624   S.citg 22625  areacarea 23930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-sum 13802  df-itg 22630  df-area 23931

This theorem is referenced by:  areambl  23933  areass  23934  areaf  23936  areacirc  32082  arearect  36145  areaquad  36146