Theorem salgenval 46507

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 46499	. . 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 4872	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ∪ 𝑥 = ∪ 𝑋)
4	3	eqeq2d 2745	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∪ 𝑠 = ∪ 𝑥 ↔ ∪ 𝑠 = ∪ 𝑋))
5		sseq1 3957	. . . . . 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 3404	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)} = {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
8	7	inteqd 4905	. . 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 3459	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
11		uniexg 7683	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑋 ∈ V)
12		pwsal 46501	. . . . . . 7 ⊢ (∪ 𝑋 ∈ V → 𝒫 ∪ 𝑋 ∈ SAlg)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝒫 ∪ 𝑋 ∈ SAlg)
14		unipw 5396	. . . . . . 7 ⊢ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋
15	14	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋)
16		pwuni 4899	. . . . . . 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 4872	. . . . . . . 8 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → ∪ 𝑠 = ∪ 𝒫 ∪ 𝑋)
20	19	eqeq1d 2736	. . . . . . 7 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋))
21		sseq2 3958	. . . . . . 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 3644	. . . . 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 4292	. . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
26		intex 5287	. . 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 6946	1 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  {crab 3397  Vcvv 3438   ⊆ wss 3899  ∅c0 4283  𝒫 cpw 4552  ∪ cuni 4861  ∩ cint 4900   ↦ cmpt 5177  ‘cfv 6490  SAlgcsalg 46494  SalGencsalgen 46498

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-salg 46495  df-salgen 46499

This theorem is referenced by:  salgencl  46518  sssalgen  46521  salgenss  46522  salgenuni  46523  issalgend  46524