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Theorem elrnsiga 31380
Description: Dropping the base information off a sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
Assertion
Ref Expression
elrnsiga (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ran sigAlgebra)

Proof of Theorem elrnsiga
StepHypRef Expression
1 fvssunirn 6693 . 2 (sigAlgebra‘𝑂) ⊆ ran sigAlgebra
21sseli 3962 1 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110   cuni 4831  ran crn 5550  cfv 6349  sigAlgebracsiga 31362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-cnv 5557  df-dm 5559  df-rn 5560  df-iota 6308  df-fv 6357
This theorem is referenced by:  sgsiga  31396  sigapisys  31409  sigaldsys  31413  brsiga  31437  sxsiga  31445  measinb2  31477  pwcntmeas  31481  ddemeas  31490  cnmbfm  31516  elmbfmvol2  31520  mbfmcnt  31521  br2base  31522  dya2iocbrsiga  31528  dya2icobrsiga  31529  sxbrsiga  31543  omsmeas  31576  isrrvv  31696  rrvadd  31705  rrvmulc  31706  dstrvprob  31724
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