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Theorem sigagenss 31516
 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 5194 . . . 4 ((𝐴𝑆𝑆 ∈ (sigAlgebra‘ 𝐴)) → 𝐴 ∈ V)
21ancoms 462 . . 3 ((𝑆 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴𝑆) → 𝐴 ∈ V)
3 sigagenval 31507 . . 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 3944 . . 3 (𝑠 = 𝑆 → (𝐴𝑠𝐴𝑆))
65intminss 4867 . 2 ((𝑆 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴𝑆) → {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠} ⊆ 𝑆)
74, 6eqsstrd 3956 1 ((𝑆 ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴𝑆) → (sigaGen‘𝐴) ⊆ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  {crab 3113  Vcvv 3444   ⊆ wss 3884  ∪ cuni 4803  ∩ cint 4841  ‘cfv 6328  sigAlgebracsiga 31475  sigaGencsigagen 31505 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-siga 31476  df-sigagen 31506 This theorem is referenced by:  sigagenss2  31517  sigagenid  31518  imambfm  31628
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