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Theorem sigagenss2 33994
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 33985 . . . 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 6895 . . 3 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigAlgebra‘ 𝐴) = (sigAlgebra‘ 𝐵))
52, 4eleqtrrd 2829 . 2 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐴))
6 simp2 1134 . 2 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → 𝐴 ⊆ (sigaGen‘𝐵))
7 sigagenss 33993 . 2 (((sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ (sigaGen‘𝐵)) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
85, 6, 7syl2anc 582 1 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1534  wcel 2099  wss 3947   cuni 4906  cfv 6544  sigAlgebracsiga 33952  sigaGencsigagen 33982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6496  df-fun 6546  df-fv 6552  df-siga 33953  df-sigagen 33983
This theorem is referenced by:  sxbrsigalem3  34117  sxbrsigalem1  34130  sxbrsigalem2  34131  sxbrsigalem4  34132  sxbrsigalem5  34133  sxbrsiga  34135
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