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Theorem sigagenss 30810
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 5041 . . . 4 ((𝐴𝑆𝑆 ∈ (sigAlgebra‘ 𝐴)) → 𝐴 ∈ V)
21ancoms 452 . . 3 ((𝑆 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴𝑆) → 𝐴 ∈ V)
3 sigagenval 30801 . . 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 3846 . . 3 (𝑠 = 𝑆 → (𝐴𝑠𝐴𝑆))
65intminss 4736 . 2 ((𝑆 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴𝑆) → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ 𝑆)
74, 6eqsstrd 3858 1 ((𝑆 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴𝑆) → (sigaGen‘𝐴) ⊆ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  {crab 3094  Vcvv 3398  wss 3792   cuni 4671   cint 4710  cfv 6135  sigAlgebracsiga 30768  sigaGencsigagen 30799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-siga 30769  df-sigagen 30800
This theorem is referenced by:  sigagenss2  30811  sigagenid  30812  imambfm  30922
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