Theorem elrnsiga 34425

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 6900	. 2 ⊢ (sigAlgebra‘𝑂) ⊆ ∪ ran sigAlgebra
2	1	sseli 3934	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∈ wcel 2144  ∪ cuni 4867  ran crn 5650  ‘cfv 6523  sigAlgebracsiga 34407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-cnv 5657  df-dm 5659  df-rn 5660  df-iota 6479  df-fv 6531

This theorem is referenced by:  sgsiga  34441  sigapisys  34454  sigaldsys  34458  brsiga  34482  sxsiga  34490  measinb2  34522  pwcntmeas  34526  ddemeas  34535  cnmbfm  34562  elmbfmvol2  34566  mbfmcnt  34567  br2base  34568  dya2iocbrsiga  34574  dya2icobrsiga  34575  sxbrsiga  34589  omsmeas  34622  isrrvv  34742  rrvadd  34751  rrvmulc  34752  dstrvprob  34771