Theorem elrnsiga 33193

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

Step	Hyp	Ref	Expression
1		fvssunirn 6924	. 2 ⊢ (sigAlgebra‘𝑂) ⊆ ∪ ran sigAlgebra
2	1	sseli 3978	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∈ wcel 2106  ∪ cuni 4908  ran crn 5677  ‘cfv 6543  sigAlgebracsiga 33175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-iota 6495  df-fv 6551

This theorem is referenced by:  sgsiga  33209  sigapisys  33222  sigaldsys  33226  brsiga  33250  sxsiga  33258  measinb2  33290  pwcntmeas  33294  ddemeas  33303  cnmbfm  33331  elmbfmvol2  33335  mbfmcnt  33336  br2base  33337  dya2iocbrsiga  33343  dya2icobrsiga  33344  sxbrsiga  33358  omsmeas  33391  isrrvv  33511  rrvadd  33520  rrvmulc  33521  dstrvprob  33539