Theorem elrnsiga 34276

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 6863	. 2 ⊢ (sigAlgebra‘𝑂) ⊆ ∪ ran sigAlgebra
2	1	sseli 3918	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∈ wcel 2114  ∪ cuni 4851  ran crn 5623  ‘cfv 6490  sigAlgebracsiga 34258

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 5231  ax-nul 5241  ax-pr 5368

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-cnv 5630  df-dm 5632  df-rn 5633  df-iota 6446  df-fv 6498

This theorem is referenced by:  sgsiga  34292  sigapisys  34305  sigaldsys  34309  brsiga  34333  sxsiga  34341  measinb2  34373  pwcntmeas  34377  ddemeas  34386  cnmbfm  34413  elmbfmvol2  34417  mbfmcnt  34418  br2base  34419  dya2iocbrsiga  34425  dya2icobrsiga  34426  sxbrsiga  34440  omsmeas  34473  isrrvv  34593  rrvadd  34602  rrvmulc  34603  dstrvprob  34622