Theorem sigagenval 34176

Description: Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)

Assertion

Ref	Expression
sigagenval	⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})

Distinct variable group:   𝐴,𝑠

Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem sigagenval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sigagen 34175	. . 3 ⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠})
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠}))
3		unieq 4899	. . . . . 6 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
4	3	fveq2d 6885	. . . . 5 ⊢ (𝑥 = 𝐴 → (sigAlgebra‘∪ 𝑥) = (sigAlgebra‘∪ 𝐴))
5		sseq1 3989	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝑠))
6	4, 5	rabeqbidv 3439	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
7	6	inteqd 4932	. . 3 ⊢ (𝑥 = 𝐴 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
8	7	adantl 481	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = 𝐴) → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
9		elex 3485	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
10		uniexg 7739	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
11		pwsiga 34166	. . . . . . 7 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴))
12	10, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴))
13		pwuni 4926	. . . . . 6 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
14	12, 13	jctir 520	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴))
15		sseq2 3990	. . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝐴 → (𝐴 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝒫 ∪ 𝐴))
16	15	elrab 3676	. . . . 5 ⊢ (𝒫 ∪ 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ↔ (𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴))
17	14, 16	sylibr 234	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
18	17	ne0d 4322	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅)
19		intex 5319	. . 3 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
20	18, 19	sylib 218	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
21	2, 8, 9, 20	fvmptd 6998	1 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  {crab 3420  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  ∪ cuni 4888  ∩ cint 4927   ↦ cmpt 5206  ‘cfv 6536  sigAlgebracsiga 34144  sigaGencsigagen 34174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-siga 34145  df-sigagen 34175

This theorem is referenced by:  sigagensiga  34177  sssigagen  34181  sigagenss  34185