Theorem sigagensiga 34154

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 34153	. 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
2		fvex 6835	. . . . 5 ⊢ (sigaGen‘𝐴) ∈ V
3	1, 2	eqeltrrdi 2840	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
4		intex 5280	. . . 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 4027	. . . . 5 ⊢ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ (sigAlgebra‘∪ 𝐴)
7	6	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ (sigAlgebra‘∪ 𝐴))
8		fvex 6835	. . . . 5 ⊢ (sigAlgebra‘∪ 𝐴) ∈ V
9	8	elpw2 5270	. . . 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 34150	. . 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 2831	1 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2111   ≠ wne 2928  {crab 3395  Vcvv 3436   ⊆ wss 3897  ∅c0 4280  𝒫 cpw 4547  ∪ cuni 4856  ∩ cint 4895  ‘cfv 6481  sigAlgebracsiga 34121  sigaGencsigagen 34151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-siga 34122  df-sigagen 34152

This theorem is referenced by:  sgsiga  34155  unisg  34156  sigagenss2  34163  brsiga  34196  brsigarn  34197  cldssbrsiga  34200  sxsiga  34204  cnmbfm  34276  sxbrsiga  34303