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Theorem unisg 26538
Description: The sigma algebra generated by a collection  A is a sigma algebra on  U. A. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
unisg  |-  ( A  e.  V  ->  U. (sigaGen `  A )  =  U. A )

Proof of Theorem unisg
StepHypRef Expression
1 sigagensiga 26536 . . . 4  |-  ( A  e.  V  ->  (sigaGen `  A )  e.  (sigAlgebra ` 
U. A ) )
2 issgon 26518 . . . 4  |-  ( (sigaGen `  A )  e.  (sigAlgebra ` 
U. A )  <->  ( (sigaGen `  A )  e.  U. ran sigAlgebra  /\  U. A  =  U. (sigaGen `  A ) ) )
31, 2sylib 196 . . 3  |-  ( A  e.  V  ->  (
(sigaGen `  A )  e. 
U. ran sigAlgebra  /\  U. A  =  U. (sigaGen `  A
) ) )
43simprd 463 . 2  |-  ( A  e.  V  ->  U. A  =  U. (sigaGen `  A
) )
54eqcomd 2443 1  |-  ( A  e.  V  ->  U. (sigaGen `  A )  =  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   U.cuni 4086   ran crn 4836   ` cfv 5413  sigAlgebracsiga 26502  sigaGencsigagen 26533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-int 4124  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421  df-siga 26503  df-sigagen 26534
This theorem is referenced by:  unibrsiga  26552  sxsigon  26558  imambfm  26629  cnmbfm  26630  sibf0  26672  sibff  26674  sibfof  26678  sitgclg  26680  orvcval4  26795
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