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.

Step	Hyp	Ref	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 )
3	1, 2	dmmpti 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 ) }
4	3	eleq2i 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 } ) )
6	5	eleq1d 2536	. . . . 5 |- ( t = A -> ( ( t " { x } ) e. ( `' vol " RR ) <-> ( A " { x } ) e. ( `' vol " RR ) ) )
7	6	ralbidv 2906	. . . 4 |- ( t = A -> ( A. x e. RR ( t " { x } ) e. ( `' vol " RR ) <-> A. x e. RR ( A " { x } ) e. ( `' vol " RR ) ) )
8	5	fveq2d 5876	. . . . . 6 |- ( t = A -> ( vol ` ( t " { x } ) ) = ( vol ` ( A " { x } ) ) )
9	8	mpteq2dv 4540	. . . . 5 |- ( t = A -> ( x e. RR |-> ( vol ` ( t " { x } ) ) ) = ( x e. RR |-> ( vol ` ( A " { x } ) ) ) )
10	9	eleq1d 2536	. . . 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 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 ) ) )
12	11	elrab 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
14	13, 13	xpex 6599	. . . . 5 |- ( RR X. RR ) e. _V
15	14	elpw2 4617	. . . 4 |- ( A e. ~P ( RR X. RR ) <-> A C_ ( RR X. RR ) )
16	15	anbi1i 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 ) ) )
18	16, 17	bitr4i 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 ) )
19	4, 12, 18	3bitri 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 ) )

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