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

Theorem unisg 32806
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 32804 . . . 4 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
2 issgon 32786 . . . 4 ((sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴) ↔ ((sigaGen‘𝐴) ∈ ran sigAlgebra ∧ 𝐴 = (sigaGen‘𝐴)))
31, 2sylib 217 . . 3 (𝐴𝑉 → ((sigaGen‘𝐴) ∈ ran sigAlgebra ∧ 𝐴 = (sigaGen‘𝐴)))
43simprd 497 . 2 (𝐴𝑉 𝐴 = (sigaGen‘𝐴))
54eqcomd 2739 1 (𝐴𝑉 (sigaGen‘𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107   cuni 4869  ran crn 5638  cfv 6500  sigAlgebracsiga 32771  sigaGencsigagen 32801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-siga 32772  df-sigagen 32802
This theorem is referenced by:  unibrsiga  32849  sxsigon  32855  imambfm  32926  cnmbfm  32927  sibf0  32998  sibff  33000  sibfof  33004  sitgclg  33006  orvcval4  33124
  Copyright terms: Public domain W3C validator