Theorem sigagensiga 34304

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 34303	. 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
2		fvex 6848	. . . . 5 ⊢ (sigaGen‘𝐴) ∈ V
3	1, 2	eqeltrrdi 2846	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
4		intex 5282	. . . 4 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
5	3, 4	sylibr 234	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅)
6		ssrab2 4021	. . . . 5 ⊢ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ (sigAlgebra‘∪ 𝐴)
7	6	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ (sigAlgebra‘∪ 𝐴))
8		fvex 6848	. . . . 5 ⊢ (sigAlgebra‘∪ 𝐴) ∈ V
9	8	elpw2 5272	. . . 4 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ 𝒫 (sigAlgebra‘∪ 𝐴) ↔ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ (sigAlgebra‘∪ 𝐴))
10	7, 9	sylibr 234	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ 𝒫 (sigAlgebra‘∪ 𝐴))
11		insiga 34300	. . 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 585	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ (sigAlgebra‘∪ 𝐴))
13	1, 12	eqeltrd 2837	1 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2933  {crab 3390  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851  ∩ cint 4890  ‘cfv 6493  sigAlgebracsiga 34271  sigaGencsigagen 34301

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 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

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 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  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 34272  df-sigagen 34302

This theorem is referenced by:  sgsiga  34305  unisg  34306  sigagenss2  34313  brsiga  34346  brsigarn  34347  cldssbrsiga  34350  sxsiga  34354  cnmbfm  34426  sxbrsiga  34453