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Theorem sigagenss2 28811
Description: Sufficient condition for inclusion of sigma algebra. This is used to prove equality of sigma algebra. (Contributed by Thierry Arnoux, 10-Oct-2017.)
Assertion
Ref Expression
sigagenss2  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  (sigaGen `  A )  C_  (sigaGen `  B ) )

Proof of Theorem sigagenss2
StepHypRef Expression
1 sigagensiga 28802 . . . 4  |-  ( B  e.  V  ->  (sigaGen `  B )  e.  (sigAlgebra ` 
U. B ) )
213ad2ant3 1028 . . 3  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  (sigaGen `  B )  e.  (sigAlgebra ` 
U. B ) )
3 simp1 1005 . . . 4  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  U. A  =  U. B )
43fveq2d 5885 . . 3  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  (sigAlgebra ` 
U. A )  =  (sigAlgebra `  U. B ) )
52, 4eleqtrrd 2520 . 2  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  (sigaGen `  B )  e.  (sigAlgebra ` 
U. A ) )
6 simp2 1006 . 2  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  A  C_  (sigaGen `  B )
)
7 sigagenss 28810 . 2  |-  ( ( (sigaGen `  B )  e.  (sigAlgebra `  U. A )  /\  A  C_  (sigaGen `  B ) )  -> 
(sigaGen `  A )  C_  (sigaGen `  B ) )
85, 6, 7syl2anc 665 1  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  (sigaGen `  A )  C_  (sigaGen `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870    C_ wss 3442   U.cuni 4222   ` cfv 5601  sigAlgebracsiga 28768  sigaGencsigagen 28799
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-iota 5565  df-fun 5603  df-fv 5609  df-siga 28769  df-sigagen 28800
This theorem is referenced by:  sxbrsigalem3  28933  sxbrsigalem1  28946  sxbrsigalem2  28947  sxbrsigalem4  28948  sxbrsigalem5  28949  sxbrsiga  28951
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