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

Theorem measbasedom 34204
Description: The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measbasedom (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))

Proof of Theorem measbasedom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isrnmeas 34202 . . . 4 (𝑀 ran measures → (dom 𝑀 ran sigAlgebra ∧ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
21simprd 495 . . 3 (𝑀 ran measures → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))
3 dmmeas 34203 . . . 4 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
4 ismeas 34201 . . . 4 (dom 𝑀 ran sigAlgebra → (𝑀 ∈ (measures‘dom 𝑀) ↔ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
53, 4syl 17 . . 3 (𝑀 ran measures → (𝑀 ∈ (measures‘dom 𝑀) ↔ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
62, 5mpbird 257 . 2 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
7 elfvunirn 6937 . 2 (𝑀 ∈ (measures‘dom 𝑀) → 𝑀 ran measures)
86, 7impbii 209 1 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  c0 4332  𝒫 cpw 4599   cuni 4906  Disj wdisj 5109   class class class wbr 5142  dom cdm 5684  ran crn 5685  wf 6556  cfv 6560  (class class class)co 7432  ωcom 7888  cdom 8984  0cc0 11156  +∞cpnf 11293  [,]cicc 13391  Σ*cesum 34029  sigAlgebracsiga 34110  measurescmeas 34197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-esum 34030  df-meas 34198
This theorem is referenced by:  truae  34245  aean  34246  sibfinima  34342  sibfof  34343  domprobmeas  34413  probmeasd  34426  probfinmeasb  34431  probfinmeasbALTV  34432  probmeasb  34433  dstrvprob  34475
  Copyright terms: Public domain W3C validator