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Theorem measbasedom 32841
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 32839 . . . 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 497 . . 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 32840 . . . 4 (𝑀 ∈ βˆͺ ran measures β†’ dom 𝑀 ∈ βˆͺ ran sigAlgebra)
4 ismeas 32838 . . . 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 6879 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆ…c0 4287  π’« cpw 4565  βˆͺ cuni 4870  Disj wdisj 5075   class class class wbr 5110  dom cdm 5638  ran crn 5639  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  Ο‰com 7807   β‰Ό cdom 8888  0cc0 11058  +∞cpnf 11193  [,]cicc 13274  Ξ£*cesum 32666  sigAlgebracsiga 32747  measurescmeas 32834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-esum 32667  df-meas 32835
This theorem is referenced by:  truae  32882  aean  32883  sibfinima  32979  sibfof  32980  domprobmeas  33050  probmeasd  33063  probfinmeasb  33068  probfinmeasbALTV  33069  probmeasb  33070  dstrvprob  33111
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