Theorem elrnsiga 33119

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 33101

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  33135  sigapisys  33148  sigaldsys  33152  brsiga  33176  sxsiga  33184  measinb2  33216  pwcntmeas  33220  ddemeas  33229  cnmbfm  33257  elmbfmvol2  33261  mbfmcnt  33262  br2base  33263  dya2iocbrsiga  33269  dya2icobrsiga  33270  sxbrsiga  33284  omsmeas  33317  isrrvv  33437  rrvadd  33446  rrvmulc  33447  dstrvprob  33465