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Theorem sigagenid 31518
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 31491 . . 3 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
2 ssid 3940 . . 3 𝑆𝑆
3 sigagenss 31516 . . 3 ((𝑆 ∈ (sigAlgebra‘ 𝑆) ∧ 𝑆𝑆) → (sigaGen‘𝑆) ⊆ 𝑆)
41, 2, 3sylancl 589 . 2 (𝑆 ran sigAlgebra → (sigaGen‘𝑆) ⊆ 𝑆)
5 sssigagen 31512 . 2 (𝑆 ran sigAlgebra → 𝑆 ⊆ (sigaGen‘𝑆))
64, 5eqssd 3935 1 (𝑆 ran sigAlgebra → (sigaGen‘𝑆) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  wss 3884   cuni 4803  ran crn 5524  cfv 6328  sigAlgebracsiga 31475  sigaGencsigagen 31505
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-siga 31476  df-sigagen 31506
This theorem is referenced by: (None)
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