Theorem sigagensiga 34325

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 34324	. 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
2		fvex 6840	. . . . 5 ⊢ (sigaGen‘𝐴) ∈ V
3	1, 2	eqeltrrdi 2848	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
4		intex 5272	. . . 4 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
5	3, 4	sylibr 235	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅)
6		ssrab2 4011	. . . . 5 ⊢ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ (sigAlgebra‘∪ 𝐴)
7	6	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ (sigAlgebra‘∪ 𝐴))
8		fvex 6840	. . . . 5 ⊢ (sigAlgebra‘∪ 𝐴) ∈ V
9	8	elpw2 5262	. . . 4 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ 𝒫 (sigAlgebra‘∪ 𝐴) ↔ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ⊆ (sigAlgebra‘∪ 𝐴))
10	7, 9	sylibr 235	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ 𝒫 (sigAlgebra‘∪ 𝐴))
11		insiga 34321	. . 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 590	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ (sigAlgebra‘∪ 𝐴))
13	1, 12	eqeltrd 2839	1 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2119   ≠ wne 2934  {crab 3391  Vcvv 3431   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  ∪ cuni 4838  ∩ cint 4877  ‘cfv 6485  sigAlgebracsiga 34292  sigaGencsigagen 34322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-siga 34293  df-sigagen 34323

This theorem is referenced by:  sgsiga  34326  unisg  34327  sigagenss2  34334  brsiga  34367  brsigarn  34368  cldssbrsiga  34371  sxsiga  34375  cnmbfm  34447  sxbrsiga  34474