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Theorem unisg 31684
Description: The sigma-algebra generated by a collection 𝐴 is a sigma-algebra on 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
unisg (𝐴𝑉 (sigaGen‘𝐴) = 𝐴)

Proof of Theorem unisg
StepHypRef Expression
1 sigagensiga 31682 . . . 4 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
2 issgon 31664 . . . 4 ((sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴) ↔ ((sigaGen‘𝐴) ∈ ran sigAlgebra ∧ 𝐴 = (sigaGen‘𝐴)))
31, 2sylib 221 . . 3 (𝐴𝑉 → ((sigaGen‘𝐴) ∈ ran sigAlgebra ∧ 𝐴 = (sigaGen‘𝐴)))
43simprd 499 . 2 (𝐴𝑉 𝐴 = (sigaGen‘𝐴))
54eqcomd 2745 1 (𝐴𝑉 (sigaGen‘𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114   cuni 4797  ran crn 5527  cfv 6340  sigAlgebracsiga 31649  sigaGencsigagen 31679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-int 4838  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-fv 6348  df-siga 31650  df-sigagen 31680
This theorem is referenced by:  unibrsiga  31727  sxsigon  31733  imambfm  31802  cnmbfm  31803  sibf0  31874  sibff  31876  sibfof  31880  sitgclg  31882  orvcval4  32000
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