Theorem salgenval 46770

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 46762	. . 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 4862	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ∪ 𝑥 = ∪ 𝑋)
4	3	eqeq2d 2748	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∪ 𝑠 = ∪ 𝑥 ↔ ∪ 𝑠 = ∪ 𝑋))
5		sseq1 3948	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑠))
6	4, 5	anbi12d 633	. . . . 5 ⊢ (𝑥 = 𝑋 → ((∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠) ↔ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)))
7	6	rabbidv 3397	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)} = {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
8	7	inteqd 4895	. . 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 3451	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
11		uniexg 7688	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑋 ∈ V)
12		pwsal 46764	. . . . . . 7 ⊢ (∪ 𝑋 ∈ V → 𝒫 ∪ 𝑋 ∈ SAlg)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝒫 ∪ 𝑋 ∈ SAlg)
14		unipw 5398	. . . . . . 7 ⊢ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋
15	14	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋)
16		pwuni 4889	. . . . . . 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 4862	. . . . . . . 8 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → ∪ 𝑠 = ∪ 𝒫 ∪ 𝑋)
20	19	eqeq1d 2739	. . . . . . 7 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋))
21		sseq2 3949	. . . . . . 7 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝒫 ∪ 𝑋))
22	20, 21	anbi12d 633	. . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝒫 ∪ 𝑋 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝒫 ∪ 𝑋)))
23	22	elrab 3635	. . . . 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 4283	. . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
26		intex 5282	. . 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 6950	1 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3390  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851  ∩ cint 4890   ↦ cmpt 5167  ‘cfv 6493  SAlgcsalg 46757  SalGencsalgen 46761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-salg 46758  df-salgen 46762

This theorem is referenced by:  salgencl  46781  sssalgen  46784  salgenss  46785  salgenuni  46786  issalgend  46787