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Theorem sigagenss2 27776
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 27767 . . . 4  |-  ( B  e.  V  ->  (sigaGen `  B )  e.  (sigAlgebra ` 
U. B ) )
213ad2ant3 1014 . . 3  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  (sigaGen `  B )  e.  (sigAlgebra ` 
U. B ) )
3 simp1 991 . . . 4  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  U. A  =  U. B )
43fveq2d 5861 . . 3  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  (sigAlgebra ` 
U. A )  =  (sigAlgebra `  U. B ) )
52, 4eleqtrrd 2551 . 2  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  (sigaGen `  B )  e.  (sigAlgebra ` 
U. A ) )
6 simp2 992 . 2  |-  ( ( U. A  =  U. B  /\  A  C_  (sigaGen `  B )  /\  B  e.  V )  ->  A  C_  (sigaGen `  B )
)
7 sigagenss 27775 . 2  |-  ( ( (sigaGen `  B )  e.  (sigAlgebra `  U. A )  /\  A  C_  (sigaGen `  B ) )  -> 
(sigaGen `  A )  C_  (sigaGen `  B ) )
85, 6, 7syl2anc 661 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 968    = wceq 1374    e. wcel 1762    C_ wss 3469   U.cuni 4238   ` cfv 5579  sigAlgebracsiga 27733  sigaGencsigagen 27764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-siga 27734  df-sigagen 27765
This theorem is referenced by:  sxbrsigalem3  27869  sxbrsigalem1  27882  sxbrsigalem2  27883  sxbrsigalem4  27884  sxbrsigalem5  27885  sxbrsiga  27887
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