Theorem sigagensiga 33208

Description: A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)

Assertion

Ref	Expression
sigagensiga	⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴))

Proof of Theorem sigagensiga

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sigagenval 33207	. 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
2		fvex 6904	. . . . 5 ⊢ (sigaGen‘𝐴) ∈ V
3	1, 2	eqeltrrdi 2842	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
4		intex 5337	. . . 4 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
5	3, 4	sylibr 233	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅)
6		ssrab2 4077	. . . . 5 ⊢ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ (sigAlgebra‘∪ 𝐴)
7	6	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ (sigAlgebra‘∪ 𝐴))
8		fvex 6904	. . . . 5 ⊢ (sigAlgebra‘∪ 𝐴) ∈ V
9	8	elpw2 5345	. . . 4 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ 𝒫 (sigAlgebra‘∪ 𝐴) ↔ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ (sigAlgebra‘∪ 𝐴))
10	7, 9	sylibr 233	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ 𝒫 (sigAlgebra‘∪ 𝐴))
11		insiga 33204	. . 3 ⊢ (({𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅ ∧ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ 𝒫 (sigAlgebra‘∪ 𝐴)) → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ (sigAlgebra‘∪ 𝐴))
12	5, 10, 11	syl2anc 584	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ (sigAlgebra‘∪ 𝐴))
13	1, 12	eqeltrd 2833	1 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2106   ≠ wne 2940  {crab 3432  Vcvv 3474   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  ∪ cuni 4908  ∩ cint 4950  ‘cfv 6543  sigAlgebracsiga 33175  sigaGencsigagen 33205

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-siga 33176  df-sigagen 33206

This theorem is referenced by:  sgsiga  33209  unisg  33210  sigagenss2  33217  brsiga  33250  brsigarn  33251  cldssbrsiga  33254  sxsiga  33258  cnmbfm  33331  sxbrsiga  33358