Theorem salgenval 46317

Description: The sigma-algebra generated by a set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Assertion

Ref	Expression
salgenval	⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})

Distinct variable group:   𝑋,𝑠

Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem salgenval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-salgen 46309	. . 3 ⊢ SalGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)})
2	1	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → SalGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)}))
3		unieq 4899	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ∪ 𝑥 = ∪ 𝑋)
4	3	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∪ 𝑠 = ∪ 𝑥 ↔ ∪ 𝑠 = ∪ 𝑋))
5		sseq1 3989	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑠))
6	4, 5	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝑋 → ((∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠) ↔ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)))
7	6	rabbidv 3428	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)} = {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
8	7	inteqd 4932	. . 3 ⊢ (𝑥 = 𝑋 → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)} = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
9	8	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)} = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
10		elex 3485	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
11		uniexg 7739	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑋 ∈ V)
12		pwsal 46311	. . . . . . 7 ⊢ (∪ 𝑋 ∈ V → 𝒫 ∪ 𝑋 ∈ SAlg)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝒫 ∪ 𝑋 ∈ SAlg)
14		unipw 5430	. . . . . . 7 ⊢ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋
15	14	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋)
16		pwuni 4926	. . . . . . 7 ⊢ 𝑋 ⊆ 𝒫 ∪ 𝑋
17	16	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝒫 ∪ 𝑋)
18	13, 15, 17	jca32 515	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝒫 ∪ 𝑋 ∈ SAlg ∧ (∪ 𝒫 ∪ 𝑋 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝒫 ∪ 𝑋)))
19		unieq 4899	. . . . . . . 8 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → ∪ 𝑠 = ∪ 𝒫 ∪ 𝑋)
20	19	eqeq1d 2738	. . . . . . 7 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋))
21		sseq2 3990	. . . . . . 7 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝒫 ∪ 𝑋))
22	20, 21	anbi12d 632	. . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝒫 ∪ 𝑋 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝒫 ∪ 𝑋)))
23	22	elrab 3676	. . . . 5 ⊢ (𝒫 ∪ 𝑋 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝒫 ∪ 𝑋 ∈ SAlg ∧ (∪ 𝒫 ∪ 𝑋 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝒫 ∪ 𝑋)))
24	18, 23	sylibr 234	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝒫 ∪ 𝑋 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
25	24	ne0d 4322	. . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
26		intex 5319	. . 3 ⊢ ({𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅ ↔ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ∈ V)
27	25, 26	sylib 218	. 2 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ∈ V)
28	2, 9, 10, 27	fvmptd 6998	1 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 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  SAlgcsalg 46304  SalGencsalgen 46308

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-salg 46305  df-salgen 46309

This theorem is referenced by:  salgencl  46328  sssalgen  46331  salgenss  46332  salgenuni  46333  issalgend  46334