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Theorem unisg 28373
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 28371 . . . 4  |-  ( A  e.  V  ->  (sigaGen `  A )  e.  (sigAlgebra ` 
U. A ) )
2 issgon 28353 . . . 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 461 . 2  |-  ( A  e.  V  ->  U. A  =  U. (sigaGen `  A
) )
54eqcomd 2462 1  |-  ( A  e.  V  ->  U. (sigaGen `  A )  =  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   U.cuni 4235   ran crn 4989   ` cfv 5570  sigAlgebracsiga 28337  sigaGencsigagen 28368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-siga 28338  df-sigagen 28369
This theorem is referenced by:  unibrsiga  28394  sxsigon  28400  imambfm  28470  cnmbfm  28471  sibf0  28540  sibff  28542  sibfof  28546  sitgclg  28548  orvcval4  28663
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