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Theorem unisg 24479
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 24477 . . . 4  |-  ( A  e.  V  ->  (sigaGen `  A )  e.  (sigAlgebra ` 
U. A ) )
2 issgon 24459 . . . 4  |-  ( (sigaGen `  A )  e.  (sigAlgebra ` 
U. A )  <->  ( (sigaGen `  A )  e.  U. ran sigAlgebra  /\  U. A  =  U. (sigaGen `  A ) ) )
31, 2sylib 189 . . 3  |-  ( A  e.  V  ->  (
(sigaGen `  A )  e. 
U. ran sigAlgebra  /\  U. A  =  U. (sigaGen `  A
) ) )
43simprd 450 . 2  |-  ( A  e.  V  ->  U. A  =  U. (sigaGen `  A
) )
54eqcomd 2409 1  |-  ( A  e.  V  ->  U. (sigaGen `  A )  =  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   U.cuni 3975   ran crn 4838   ` cfv 5413  sigAlgebracsiga 24443  sigaGencsigagen 24474
This theorem is referenced by:  unibrsiga  24493  sxsigon  24499  imambfm  24565  cnmbfm  24566  sibf0  24602  sibff  24604  sibfof  24607  sitgclg  24609  orvcval4  24671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-siga 24444  df-sigagen 24475
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