Theorem elrnsiga 34146

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 6859	. 2 ⊢ (sigAlgebra‘𝑂) ⊆ ∪ ran sigAlgebra
2	1	sseli 3925	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∈ wcel 2111  ∪ cuni 4858  ran crn 5620  ‘cfv 6487  sigAlgebracsiga 34128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-cnv 5627  df-dm 5629  df-rn 5630  df-iota 6443  df-fv 6495

This theorem is referenced by:  sgsiga  34162  sigapisys  34175  sigaldsys  34179  brsiga  34203  sxsiga  34211  measinb2  34243  pwcntmeas  34247  ddemeas  34256  cnmbfm  34283  elmbfmvol2  34287  mbfmcnt  34288  br2base  34289  dya2iocbrsiga  34295  dya2icobrsiga  34296  sxbrsiga  34310  omsmeas  34343  isrrvv  34463  rrvadd  34472  rrvmulc  34473  dstrvprob  34492