Theorem isi1f 23441

Description: The predicate " F is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom F e. dom S.1 to represent this concept because S.1 is the first preparation function for our final definition S. (see df-itg 23392); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)

Assertion

Ref	Expression
isi1f	|- ( F e. dom S.1 <-> ( F e. MblFn /\ ( F : RR --> RR /\ ran F e. Fin /\ ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) ) )

Proof of Theorem isi1f

Dummy variables f g x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		feq1 6026	. . 3 |- ( g = F -> ( g : RR --> RR <-> F : RR --> RR ) )
2		rneq 5351	. . . 4 |- ( g = F -> ran g = ran F )
3	2	eleq1d 2686	. . 3 |- ( g = F -> ( ran g e. Fin <-> ran F e. Fin ) )
4		cnveq 5296	. . . . . 6 |- ( g = F -> `' g = `' F )
5	4	imaeq1d 5465	. . . . 5 |- ( g = F -> ( `' g " ( RR \ { 0 } ) ) = ( `' F " ( RR \ { 0 } ) ) )
6	5	fveq2d 6195	. . . 4 |- ( g = F -> ( vol ` ( `' g " ( RR \ { 0 } ) ) ) = ( vol ` ( `' F " ( RR \ { 0 } ) ) ) )
7	6	eleq1d 2686	. . 3 |- ( g = F -> ( ( vol ` ( `' g " ( RR \ { 0 } ) ) ) e. RR <-> ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) )
8	1, 3, 7	3anbi123d 1399	. 2 |- ( g = F -> ( ( g : RR --> RR /\ ran g e. Fin /\ ( vol ` ( `' g " ( RR \ { 0 } ) ) ) e. RR ) <-> ( F : RR --> RR /\ ran F e. Fin /\ ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) ) )
9		sumex 14418	. . 3 |- sum_ x e. ( ran f \ { 0 } ) ( x x. ( vol ` ( `' f " { x } ) ) ) e. _V
10		df-itg1 23389	. . 3 |- S.1 = ( f e. { g e. MblFn | ( g : RR --> RR /\ ran g e. Fin /\ ( vol ` ( `' g " ( RR \ { 0 } ) ) ) e. RR ) } |-> sum_ x e. ( ran f \ { 0 } ) ( x x. ( vol ` ( `' f " { x } ) ) ) )
11	9, 10	dmmpti 6023	. 2 |- dom S.1 = { g e. MblFn | ( g : RR --> RR /\ ran g e. Fin /\ ( vol ` ( `' g " ( RR \ { 0 } ) ) ) e. RR ) }
12	8, 11	elrab2 3366	1 |- ( F e. dom S.1 <-> ( F e. MblFn /\ ( F : RR --> RR /\ ran F e. Fin /\ ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   {csn 4177   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   RRcr 9935   0cc0 9936   x. cmul 9941   sum_csu 14416   volcvol 23232  MblFncmbf 23383   S.1citg1 23384

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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-sum 14417  df-itg1 23389

This theorem is referenced by:  i1fmbf  23442  i1ff  23443  i1frn  23444  i1fima2  23446  i1fd  23448