Theorem elrnsiga 34262

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 6864	. 2 ⊢ (sigAlgebra‘𝑂) ⊆ ∪ ran sigAlgebra
2	1	sseli 3928	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∈ wcel 2114  ∪ cuni 4862  ran crn 5624  ‘cfv 6491  sigAlgebracsiga 34244

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 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-cnv 5631  df-dm 5633  df-rn 5634  df-iota 6447  df-fv 6499

This theorem is referenced by:  sgsiga  34278  sigapisys  34291  sigaldsys  34295  brsiga  34319  sxsiga  34327  measinb2  34359  pwcntmeas  34363  ddemeas  34372  cnmbfm  34399  elmbfmvol2  34403  mbfmcnt  34404  br2base  34405  dya2iocbrsiga  34411  dya2icobrsiga  34412  sxbrsiga  34426  omsmeas  34459  isrrvv  34579  rrvadd  34588  rrvmulc  34589  dstrvprob  34608