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Theorem sigagenid 31309
Description: The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.)
Assertion
Ref Expression
sigagenid (𝑆 ran sigAlgebra → (sigaGen‘𝑆) = 𝑆)

Proof of Theorem sigagenid
StepHypRef Expression
1 sgon 31282 . . 3 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
2 ssid 3986 . . 3 𝑆𝑆
3 sigagenss 31307 . . 3 ((𝑆 ∈ (sigAlgebra‘ 𝑆) ∧ 𝑆𝑆) → (sigaGen‘𝑆) ⊆ 𝑆)
41, 2, 3sylancl 586 . 2 (𝑆 ran sigAlgebra → (sigaGen‘𝑆) ⊆ 𝑆)
5 sssigagen 31303 . 2 (𝑆 ran sigAlgebra → 𝑆 ⊆ (sigaGen‘𝑆))
64, 5eqssd 3981 1 (𝑆 ran sigAlgebra → (sigaGen‘𝑆) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wss 3933   cuni 4830  ran crn 5549  cfv 6348  sigAlgebracsiga 31266  sigaGencsigagen 31296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-siga 31267  df-sigagen 31297
This theorem is referenced by: (None)
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