Theorem sigagenval 33815

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 33814	. . 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 4914	. . . . . 6 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
4	3	fveq2d 6895	. . . . 5 ⊢ (𝑥 = 𝐴 → (sigAlgebra‘∪ 𝑥) = (sigAlgebra‘∪ 𝐴))
5		sseq1 3998	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝑠))
6	4, 5	rabeqbidv 3437	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
7	6	inteqd 4949	. . 3 ⊢ (𝑥 = 𝐴 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
8	7	adantl 480	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = 𝐴) → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
9		elex 3482	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
10		uniexg 7742	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
11		pwsiga 33805	. . . . . . 7 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴))
12	10, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴))
13		pwuni 4943	. . . . . 6 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
14	12, 13	jctir 519	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴))
15		sseq2 3999	. . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝐴 → (𝐴 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝒫 ∪ 𝐴))
16	15	elrab 3675	. . . . 5 ⊢ (𝒫 ∪ 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ↔ (𝒫 ∪ 𝐴 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴))
17	14, 16	sylibr 233	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
18	17	ne0d 4331	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅)
19		intex 5334	. . 3 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
20	18, 19	sylib 217	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠} ∈ V)
21	2, 8, 9, 20	fvmptd 7006	1 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  {crab 3419  Vcvv 3463   ⊆ wss 3940  ∅c0 4318  𝒫 cpw 4598  ∪ cuni 4903  ∩ cint 4944   ↦ cmpt 5226  ‘cfv 6542  sigAlgebracsiga 33783  sigaGencsigagen 33813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-siga 33784  df-sigagen 33814

This theorem is referenced by:  sigagensiga  33816  sssigagen  33820  sigagenss  33824