Theorem elrnsiga 34296

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 6866	. 2 ⊢ (sigAlgebra‘𝑂) ⊆ ∪ ran sigAlgebra
2	1	sseli 3930	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∈ wcel 2114  ∪ cuni 4864  ran crn 5626  ‘cfv 6493  sigAlgebracsiga 34278

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-cnv 5633  df-dm 5635  df-rn 5636  df-iota 6449  df-fv 6501

This theorem is referenced by:  sgsiga  34312  sigapisys  34325  sigaldsys  34329  brsiga  34353  sxsiga  34361  measinb2  34393  pwcntmeas  34397  ddemeas  34406  cnmbfm  34433  elmbfmvol2  34437  mbfmcnt  34438  br2base  34439  dya2iocbrsiga  34445  dya2icobrsiga  34446  sxbrsiga  34460  omsmeas  34493  isrrvv  34613  rrvadd  34622  rrvmulc  34623  dstrvprob  34642