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Theorem sgsiga 28811
Description: A generated sigma algebra is a sigma algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypothesis
Ref Expression
sgsiga.1  |-  ( ph  ->  A  e.  V )
Assertion
Ref Expression
sgsiga  |-  ( ph  ->  (sigaGen `  A )  e.  U. ran sigAlgebra )

Proof of Theorem sgsiga
StepHypRef Expression
1 sgsiga.1 . 2  |-  ( ph  ->  A  e.  V )
2 sigagensiga 28810 . 2  |-  ( A  e.  V  ->  (sigaGen `  A )  e.  (sigAlgebra ` 
U. A ) )
3 elrnsiga 28795 . 2  |-  ( (sigaGen `  A )  e.  (sigAlgebra ` 
U. A )  -> 
(sigaGen `  A )  e. 
U. ran sigAlgebra )
41, 2, 33syl 18 1  |-  ( ph  ->  (sigaGen `  A )  e.  U. ran sigAlgebra )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   U.cuni 4222   ran crn 4855   ` cfv 5601  sigAlgebracsiga 28776  sigaGencsigagen 28807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-siga 28777  df-sigagen 28808
This theorem is referenced by:  elsigagen2  28817  cldssbrsiga  28856  mbfmbfm  28927  imambfm  28931  sxbrsigalem2  28955  sxbrsiga  28959  sibf0  29003  sibff  29005  sibfinima  29008  sibfof  29009  sitgclg  29011  orvcval4  29127  orvcoel  29128  orvccel  29129
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