Theorem elrnsiga 29516

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 6127	. 2 ⊢ (sigAlgebra‘𝑂) ⊆ ∪ ran sigAlgebra
2	1	sseli 3564	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∈ wcel 1977  ∪ cuni 4372  ran crn 5039  ‘cfv 5804  sigAlgebracsiga 29497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-iota 5768  df-fv 5812

This theorem is referenced by:  sgsiga  29532  sigapisys  29545  sigaldsys  29549  brsiga  29573  sxsiga  29581  measinb2  29613  pwcntmeas  29617  ddemeas  29626  cnmbfm  29652  elmbfmvol2  29656  mbfmcnt  29657  br2base  29658  dya2iocbrsiga  29664  dya2icobrsiga  29665  sxbrsiga  29679  omsmeas  29712  isrrvv  29832  rrvadd  29841  rrvmulc  29842  dstrvprob  29860