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Theorem sgsiga 33660
Description: A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypothesis
Ref Expression
sgsiga.1 (𝜑𝐴𝑉)
Assertion
Ref Expression
sgsiga (𝜑 → (sigaGen‘𝐴) ∈ ran sigAlgebra)

Proof of Theorem sgsiga
StepHypRef Expression
1 sgsiga.1 . 2 (𝜑𝐴𝑉)
2 sigagensiga 33659 . 2 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
3 elrnsiga 33644 . 2 ((sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴) → (sigaGen‘𝐴) ∈ ran sigAlgebra)
41, 2, 33syl 18 1 (𝜑 → (sigaGen‘𝐴) ∈ ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   cuni 4900  ran crn 5668  cfv 6534  sigAlgebracsiga 33626  sigaGencsigagen 33656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-siga 33627  df-sigagen 33657
This theorem is referenced by:  elsigagen2  33666  cldssbrsiga  33705  imambfm  33781  sxbrsigalem2  33805  sxbrsiga  33809  sibf0  33853  sibff  33855  sibfinima  33858  sibfof  33859  sitgclg  33861  orvcval4  33979  orvcoel  33980  orvccel  33981
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