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Theorem unisg 26440
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 26438 . . . 4  |-  ( A  e.  V  ->  (sigaGen `  A )  e.  (sigAlgebra ` 
U. A ) )
2 issgon 26420 . . . 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 460 . 2  |-  ( A  e.  V  ->  U. A  =  U. (sigaGen `  A
) )
54eqcomd 2438 1  |-  ( A  e.  V  ->  U. (sigaGen `  A )  =  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   U.cuni 4079   ran crn 4828   ` cfv 5406  sigAlgebracsiga 26404  sigaGencsigagen 26435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-fv 5414  df-siga 26405  df-sigagen 26436
This theorem is referenced by:  unibrsiga  26454  sxsigon  26460  imambfm  26531  cnmbfm  26532  sibf0  26568  sibff  26570  sibfof  26574  sitgclg  26576  orvcval4  26691
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