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Theorem sigagenss 34130
Description: The generated sigma-algebra is a subset of all sigma-algebras containing the generating set, i.e. the generated sigma-algebra is the smallest sigma-algebra containing the generating set, here 𝐴. (Contributed by Thierry Arnoux, 4-Jun-2017.)
Assertion
Ref Expression
sigagenss ((𝑆 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴𝑆) → (sigaGen‘𝐴) ⊆ 𝑆)

Proof of Theorem sigagenss
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ssexg 5329 . . . 4 ((𝐴𝑆𝑆 ∈ (sigAlgebra‘ 𝐴)) → 𝐴 ∈ V)
21ancoms 458 . . 3 ((𝑆 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴𝑆) → 𝐴 ∈ V)
3 sigagenval 34121 . . 3 (𝐴 ∈ V → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
42, 3syl 17 . 2 ((𝑆 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴𝑆) → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
5 sseq2 4022 . . 3 (𝑠 = 𝑆 → (𝐴𝑠𝐴𝑆))
65intminss 4979 . 2 ((𝑆 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴𝑆) → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ 𝑆)
74, 6eqsstrd 4034 1 ((𝑆 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴𝑆) → (sigaGen‘𝐴) ⊆ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  wss 3963   cuni 4912   cint 4951  cfv 6563  sigAlgebracsiga 34089  sigaGencsigagen 34119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-siga 34090  df-sigagen 34120
This theorem is referenced by:  sigagenss2  34131  sigagenid  34132  imambfm  34244
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