Theorem dfarea 24687

Description: Rewrite df-area 24683 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)

Assertion

Ref	Expression
dfarea	|- area = ( s e. dom area |-> S. RR ( vol ` ( s " { x } ) ) _d x )

Distinct variable group:   x, s

Proof of Theorem dfarea

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-area 24683	. 2 |- area = ( s e. { y e. ~P ( RR X. RR ) | ( A. x e. RR ( y " { x } ) e. ( `' vol " RR ) /\ ( x e. RR |-> ( vol ` ( y " { x } ) ) ) e. L^1 ) } |-> S. RR ( vol ` ( s " { x } ) ) _d x )
2		itgex 23537	. . . 4 |- S. RR ( vol ` ( s " { x } ) ) _d x e. _V
3	2, 1	dmmpti 6023	. . 3 |- dom area = { y e. ~P ( RR X. RR ) | ( A. x e. RR ( y " { x } ) e. ( `' vol " RR ) /\ ( x e. RR |-> ( vol ` ( y " { x } ) ) ) e. L^1 ) }
4		mpteq1 4737	. . 3 |- ( dom area = { y e. ~P ( RR X. RR ) | ( A. x e. RR ( y " { x } ) e. ( `' vol " RR ) /\ ( x e. RR |-> ( vol ` ( y " { x } ) ) ) e. L^1 ) } -> ( s e. dom area |-> S. RR ( vol ` ( s " { x } ) ) _d x ) = ( s e. { y e. ~P ( RR X. RR ) | ( A. x e. RR ( y " { x } ) e. ( `' vol " RR ) /\ ( x e. RR |-> ( vol ` ( y " { x } ) ) ) e. L^1 ) } |-> S. RR ( vol ` ( s " { x } ) ) _d x ) )
5	3, 4	ax-mp 5	. 2 |- ( s e. dom area |-> S. RR ( vol ` ( s " { x } ) ) _d x ) = ( s e. { y e. ~P ( RR X. RR ) | ( A. x e. RR ( y " { x } ) e. ( `' vol " RR ) /\ ( x e. RR |-> ( vol ` ( y " { x } ) ) ) e. L^1 ) } |-> S. RR ( vol ` ( s " { x } ) ) _d x )
6	1, 5	eqtr4i 2647	1 |- area = ( s e. dom area |-> S. RR ( vol ` ( s " { x } ) ) _d x )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   ~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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-iota 5851  df-fun 5890  df-fn 5891  df-sum 14417  df-itg 23392  df-area 24683

This theorem is referenced by:  areaf  24688  areaval  24691