Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  measbasedom Structured version   Unicode version

Theorem measbasedom 28365
Description: The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measbasedom  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )

Proof of Theorem measbasedom
Dummy variables  x  y  m  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isrnmeas 28363 . . . 4  |-  ( M  e.  U. ran measures  ->  ( dom  M  e.  U. ran sigAlgebra  /\  ( M : dom  M --> ( 0 [,] +oo )  /\  ( M `  (/) )  =  0  /\ 
A. x  e.  ~P  dom  M ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `  y ) ) ) ) )
21simprd 461 . . 3  |-  ( M  e.  U. ran measures  ->  ( M : dom  M --> ( 0 [,] +oo )  /\  ( M `  (/) )  =  0  /\  A. x  e.  ~P  dom  M ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `  y ) ) ) )
3 dmmeas 28364 . . . 4  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
4 ismeas 28362 . . . 4  |-  ( dom 
M  e.  U. ran sigAlgebra  ->  ( M  e.  (measures `  dom  M )  <->  ( M : dom  M --> ( 0 [,] +oo )  /\  ( M `  (/) )  =  0  /\  A. x  e.  ~P  dom  M ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `  y ) ) ) ) )
53, 4syl 16 . . 3  |-  ( M  e.  U. ran measures  ->  ( M  e.  (measures `  dom  M )  <->  ( M : dom  M --> ( 0 [,] +oo )  /\  ( M `  (/) )  =  0  /\  A. x  e.  ~P  dom  M ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `  y ) ) ) ) )
62, 5mpbird 232 . 2  |-  ( M  e.  U. ran measures  ->  M  e.  (measures `  dom  M ) )
7 df-meas 28359 . . . 4  |- measures  =  ( s  e.  U. ran sigAlgebra  |->  { m  |  ( m : s --> ( 0 [,] +oo )  /\  ( m `  (/) )  =  0  /\  A. x  e.  ~P  s ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) } )
87funmpt2 5546 . . 3  |-  Fun measures
9 elunirn2 27659 . . 3  |-  ( ( Fun measures  /\  M  e.  (measures `  dom  M ) )  ->  M  e.  U. ran measures )
108, 9mpan 668 . 2  |-  ( M  e.  (measures `  dom  M )  ->  M  e.  U.
ran measures )
116, 10impbii 188 1  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   {cab 2377   A.wral 2742   (/)c0 3724   ~Pcpw 3940   U.cuni 4176  Disj wdisj 4351   class class class wbr 4380   dom cdm 4926   ran crn 4927   Fun wfun 5503   -->wf 5505   ` cfv 5509  (class class class)co 6214   omcom 6617    ~<_ cdom 7451   0cc0 9421   +oocpnf 9554   [,]cicc 11471  Σ*cesum 28206  sigAlgebracsiga 28287  measurescmeas 28358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-fv 5517  df-ov 6217  df-esum 28207  df-meas 28359
This theorem is referenced by:  truae  28407  aean  28408  mbfmbfm  28421  sibfinima  28500  sibfof  28501  domprobmeas  28568  probmeasd  28581  probfinmeasbOLD  28586  probfinmeasb  28587  probmeasb  28588  dstrvprob  28629
  Copyright terms: Public domain W3C validator