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Theorem sgon 30785
Description: A sigma-algebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
Assertion
Ref Expression
sgon (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))

Proof of Theorem sgon
StepHypRef Expression
1 eqid 2777 . 2 𝑆 = 𝑆
2 issgon 30784 . . 3 (𝑆 ∈ (sigAlgebra‘ 𝑆) ↔ (𝑆 ran sigAlgebra ∧ 𝑆 = 𝑆))
32biimpri 220 . 2 ((𝑆 ran sigAlgebra ∧ 𝑆 = 𝑆) → 𝑆 ∈ (sigAlgebra‘ 𝑆))
41, 3mpan2 681 1 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2106   cuni 4671  ran crn 5356  cfv 6135  sigAlgebracsiga 30768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143  df-siga 30769
This theorem is referenced by:  elsigass  30786  isrnsigau  30788  unielsiga  30789  sigagenid  30812  1stmbfm  30920  2ndmbfm  30921  unveldomd  31076  probmeasb  31091
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