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Theorem elrnsiga 30520
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 6380 . 2 (sigAlgebra‘𝑂) ⊆ ran sigAlgebra
21sseli 3741 1 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2140   cuni 4589  ran crn 5268  cfv 6050  sigAlgebracsiga 30501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-cnv 5275  df-dm 5277  df-rn 5278  df-iota 6013  df-fv 6058
This theorem is referenced by:  sgsiga  30536  sigapisys  30549  sigaldsys  30553  brsiga  30577  sxsiga  30585  measinb2  30617  pwcntmeas  30621  ddemeas  30630  cnmbfm  30656  elmbfmvol2  30660  mbfmcnt  30661  br2base  30662  dya2iocbrsiga  30668  dya2icobrsiga  30669  sxbrsiga  30683  omsmeas  30716  isrrvv  30836  rrvadd  30845  rrvmulc  30846  dstrvprob  30864
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