Theorem sigagenss 34345

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 5254	. . . 4 ⊢ ((𝐴 ⊆ 𝑆 ∧ 𝑆 ∈ (sigAlgebra‘∪ 𝐴)) → 𝐴 ∈ V)
2	1	ancoms 460	. . 3 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → 𝐴 ∈ V)
3		sigagenval 34336	. . 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 3943	. . 3 ⊢ (𝑠 = 𝑆 → (𝐴 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝑆))
6	5	intminss 4907	. 2 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ 𝑆)
7	4, 6	eqsstrd 3951	1 ⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → (sigaGen‘𝐴) ⊆ 𝑆)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {crab 3393  Vcvv 3433   ⊆ wss 3885  ∪ cuni 4841  ∩ cint 4880  ‘cfv 6489  sigAlgebracsiga 34304  sigaGencsigagen 34334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-siga 34305  df-sigagen 34335

This theorem is referenced by:  sigagenss2  34346  sigagenid  34347  imambfm  34458