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Theorem sgsiga 33761
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 33760 . 2 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
3 elrnsiga 33745 . 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 2099   cuni 4908  ran crn 5679  cfv 6548  sigAlgebracsiga 33727  sigaGencsigagen 33757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-siga 33728  df-sigagen 33758
This theorem is referenced by:  elsigagen2  33767  cldssbrsiga  33806  imambfm  33882  sxbrsigalem2  33906  sxbrsiga  33910  sibf0  33954  sibff  33956  sibfinima  33959  sibfof  33960  sitgclg  33962  orvcval4  34080  orvcoel  34081  orvccel  34082
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