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Theorem sgsiga 32741
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 32740 . 2 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
3 elrnsiga 32725 . 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 2106   cuni 4865  ran crn 5634  cfv 6496  sigAlgebracsiga 32707  sigaGencsigagen 32737
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fv 6504  df-siga 32708  df-sigagen 32738
This theorem is referenced by:  elsigagen2  32747  cldssbrsiga  32786  imambfm  32862  sxbrsigalem2  32886  sxbrsiga  32890  sibf0  32934  sibff  32936  sibfinima  32939  sibfof  32940  sitgclg  32942  orvcval4  33060  orvcoel  33061  orvccel  33062
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