Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sigagenid Structured version   Visualization version   GIF version

Theorem sigagenid 32115
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 32088 . . 3 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
2 ssid 3948 . . 3 𝑆𝑆
3 sigagenss 32113 . . 3 ((𝑆 ∈ (sigAlgebra‘ 𝑆) ∧ 𝑆𝑆) → (sigaGen‘𝑆) ⊆ 𝑆)
41, 2, 3sylancl 586 . 2 (𝑆 ran sigAlgebra → (sigaGen‘𝑆) ⊆ 𝑆)
5 sssigagen 32109 . 2 (𝑆 ran sigAlgebra → 𝑆 ⊆ (sigaGen‘𝑆))
64, 5eqssd 3943 1 (𝑆 ran sigAlgebra → (sigaGen‘𝑆) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  wss 3892   cuni 4845  ran crn 5591  cfv 6432  sigAlgebracsiga 32072  sigaGencsigagen 32102
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 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-fv 6440  df-siga 32073  df-sigagen 32103
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator