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Theorem sigagenid 28981
Description: The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.)
Assertion
Ref Expression
sigagenid  |-  ( S  e.  U. ran sigAlgebra  ->  (sigaGen `  S )  =  S )

Proof of Theorem sigagenid
StepHypRef Expression
1 sgon 28954 . . 3  |-  ( S  e.  U. ran sigAlgebra  ->  S  e.  (sigAlgebra `  U. S ) )
2 ssid 3483 . . 3  |-  S  C_  S
3 sigagenss 28979 . . 3  |-  ( ( S  e.  (sigAlgebra `  U. S )  /\  S  C_  S )  ->  (sigaGen `  S )  C_  S
)
41, 2, 3sylancl 666 . 2  |-  ( S  e.  U. ran sigAlgebra  ->  (sigaGen `  S )  C_  S
)
5 sssigagen 28975 . 2  |-  ( S  e.  U. ran sigAlgebra  ->  S  C_  (sigaGen `  S )
)
64, 5eqssd 3481 1  |-  ( S  e.  U. ran sigAlgebra  ->  (sigaGen `  S )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872    C_ wss 3436   U.cuni 4219   ran crn 4854   ` cfv 5601  sigAlgebracsiga 28937  sigaGencsigagen 28968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-int 4256  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-siga 28938  df-sigagen 28969
This theorem is referenced by: (None)
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