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 30495
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 𝑚 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isrnmeas 30493 . . . 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 482 . . 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 30494 . . . 4 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
4 ismeas 30492 . . . 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 247 . 2 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
7 df-meas 30489 . . . 4 measures = (𝑠 ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))})
87funmpt2 6040 . . 3 Fun measures
9 elunirn2 29681 . . 3 ((Fun measures ∧ 𝑀 ∈ (measures‘dom 𝑀)) → 𝑀 ran measures)
108, 9mpan 708 . 2 (𝑀 ∈ (measures‘dom 𝑀) → 𝑀 ran measures)
116, 10impbii 199 1 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  {cab 2710  wral 3014  c0 4023  𝒫 cpw 4266   cuni 4544  Disj wdisj 4728   class class class wbr 4760  dom cdm 5218  ran crn 5219  Fun wfun 5995  wf 5997  cfv 6001  (class class class)co 6765  ωcom 7182  cdom 8070  0cc0 10049  +∞cpnf 10184  [,]cicc 12292  Σ*cesum 30319  sigAlgebracsiga 30400  measurescmeas 30488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fv 6009  df-ov 6768  df-esum 30320  df-meas 30489
This theorem is referenced by:  truae  30536  aean  30537  mbfmbfm  30550  sibfinima  30631  sibfof  30632  domprobmeas  30702  probmeasd  30715  probfinmeasbOLD  30720  probfinmeasb  30721  probmeasb  30722  dstrvprob  30763
  Copyright terms: Public domain W3C validator