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Theorem elrnsiga 30012
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 6184 . 2 (sigAlgebra‘𝑂) ⊆ ran sigAlgebra
21sseli 3584 1 (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987   cuni 4409  ran crn 5085  cfv 5857  sigAlgebracsiga 29993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-cnv 5092  df-dm 5094  df-rn 5095  df-iota 5820  df-fv 5865
This theorem is referenced by:  sgsiga  30028  sigapisys  30041  sigaldsys  30045  brsiga  30069  sxsiga  30077  measinb2  30109  pwcntmeas  30113  ddemeas  30122  cnmbfm  30148  elmbfmvol2  30152  mbfmcnt  30153  br2base  30154  dya2iocbrsiga  30160  dya2icobrsiga  30161  sxbrsiga  30175  omsmeas  30208  isrrvv  30328  rrvadd  30337  rrvmulc  30338  dstrvprob  30356
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