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Theorem measbasedom 33744
Description: The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measbasedom (𝑀 ∈ βˆͺ ran measures ↔ 𝑀 ∈ (measuresβ€˜dom 𝑀))

Proof of Theorem measbasedom
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isrnmeas 33742 . . . 4 (𝑀 ∈ βˆͺ ran measures β†’ (dom 𝑀 ∈ βˆͺ ran sigAlgebra ∧ (𝑀:dom π‘€βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 dom 𝑀((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘€β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘€β€˜π‘¦)))))
21simprd 495 . . 3 (𝑀 ∈ βˆͺ ran measures β†’ (𝑀:dom π‘€βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 dom 𝑀((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘€β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘€β€˜π‘¦))))
3 dmmeas 33743 . . . 4 (𝑀 ∈ βˆͺ ran measures β†’ dom 𝑀 ∈ βˆͺ ran sigAlgebra)
4 ismeas 33741 . . . 4 (dom 𝑀 ∈ βˆͺ ran sigAlgebra β†’ (𝑀 ∈ (measuresβ€˜dom 𝑀) ↔ (𝑀:dom π‘€βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 dom 𝑀((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘€β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘€β€˜π‘¦)))))
53, 4syl 17 . . 3 (𝑀 ∈ βˆͺ ran measures β†’ (𝑀 ∈ (measuresβ€˜dom 𝑀) ↔ (𝑀:dom π‘€βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 dom 𝑀((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘€β€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘€β€˜π‘¦)))))
62, 5mpbird 257 . 2 (𝑀 ∈ βˆͺ ran measures β†’ 𝑀 ∈ (measuresβ€˜dom 𝑀))
7 elfvunirn 6923 . 2 (𝑀 ∈ (measuresβ€˜dom 𝑀) β†’ 𝑀 ∈ βˆͺ ran measures)
86, 7impbii 208 1 (𝑀 ∈ βˆͺ ran measures ↔ 𝑀 ∈ (measuresβ€˜dom 𝑀))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  βˆ…c0 4318  π’« cpw 4598  βˆͺ cuni 4903  Disj wdisj 5107   class class class wbr 5142  dom cdm 5672  ran crn 5673  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414  Ο‰com 7862   β‰Ό cdom 8951  0cc0 11124  +∞cpnf 11261  [,]cicc 13345  Ξ£*cesum 33569  sigAlgebracsiga 33650  measurescmeas 33737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-esum 33570  df-meas 33738
This theorem is referenced by:  truae  33785  aean  33786  sibfinima  33882  sibfof  33883  domprobmeas  33953  probmeasd  33966  probfinmeasb  33971  probfinmeasbALTV  33972  probmeasb  33973  dstrvprob  34014
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