Theorem sigagenss 34308

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) ∧ 𝐴 ⊆ 𝑆) → (sigaGen‘𝐴) ⊆ 𝑆)

Proof of Theorem sigagenss

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssexg 5269	. . . 4 ⊢ ((𝐴 ⊆ 𝑆 ∧ 𝑆 ∈ (sigAlgebra‘∪ 𝐴)) → 𝐴 ∈ V)
2	1	ancoms 458	. . 3 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → 𝐴 ∈ V)
3		sigagenval 34299	. . 3 ⊢ (𝐴 ∈ V → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
4	2, 3	syl 17	. 2 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
5		sseq2 3961	. . 3 ⊢ (𝑠 = 𝑆 → (𝐴 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝑆))
6	5	intminss 4930	. 2 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ 𝑆)
7	4, 6	eqsstrd 3969	1 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → (sigaGen‘𝐴) ⊆ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3400  Vcvv 3441   ⊆ wss 3902  ∪ cuni 4864  ∩ cint 4903  ‘cfv 6493  sigAlgebracsiga 34267  sigaGencsigagen 34297

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-siga 34268  df-sigagen 34298

This theorem is referenced by:  sigagenss2  34309  sigagenid  34310  imambfm  34421