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Theorem sigagenss2 31830
Description: Sufficient condition for inclusion of sigma-algebras. This is used to prove equality of sigma-algebras. (Contributed by Thierry Arnoux, 10-Oct-2017.)
Assertion
Ref Expression
sigagenss2 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))

Proof of Theorem sigagenss2
StepHypRef Expression
1 sigagensiga 31821 . . . 4 (𝐵𝑉 → (sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐵))
213ad2ant3 1137 . . 3 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐵))
3 simp1 1138 . . . 4 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → 𝐴 = 𝐵)
43fveq2d 6721 . . 3 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigAlgebra‘ 𝐴) = (sigAlgebra‘ 𝐵))
52, 4eleqtrrd 2841 . 2 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐴))
6 simp2 1139 . 2 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → 𝐴 ⊆ (sigaGen‘𝐵))
7 sigagenss 31829 . 2 (((sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ (sigaGen‘𝐵)) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
85, 6, 7syl2anc 587 1 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  wcel 2110  wss 3866   cuni 4819  cfv 6380  sigAlgebracsiga 31788  sigaGencsigagen 31818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-siga 31789  df-sigagen 31819
This theorem is referenced by:  sxbrsigalem3  31951  sxbrsigalem1  31964  sxbrsigalem2  31965  sxbrsigalem4  31966  sxbrsigalem5  31967  sxbrsiga  31969
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