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Theorem sigagenss2 33678
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 33669 . . . 4 (𝐵𝑉 → (sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐵))
213ad2ant3 1132 . . 3 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐵))
3 simp1 1133 . . . 4 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → 𝐴 = 𝐵)
43fveq2d 6888 . . 3 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigAlgebra‘ 𝐴) = (sigAlgebra‘ 𝐵))
52, 4eleqtrrd 2830 . 2 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐴))
6 simp2 1134 . 2 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → 𝐴 ⊆ (sigaGen‘𝐵))
7 sigagenss 33677 . 2 (((sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ (sigaGen‘𝐵)) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
85, 6, 7syl2anc 583 1 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  wss 3943   cuni 4902  cfv 6536  sigAlgebracsiga 33636  sigaGencsigagen 33666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-siga 33637  df-sigagen 33667
This theorem is referenced by:  sxbrsigalem3  33801  sxbrsigalem1  33814  sxbrsigalem2  33815  sxbrsigalem4  33816  sxbrsigalem5  33817  sxbrsiga  33819
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