Theorem salgenval 46312

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 46304	. . 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 4878	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ∪ 𝑥 = ∪ 𝑋)
4	3	eqeq2d 2740	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∪ 𝑠 = ∪ 𝑥 ↔ ∪ 𝑠 = ∪ 𝑋))
5		sseq1 3969	. . . . . 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 3410	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)} = {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
8	7	inteqd 4911	. . 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 3465	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
11		uniexg 7696	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑋 ∈ V)
12		pwsal 46306	. . . . . . 7 ⊢ (∪ 𝑋 ∈ V → 𝒫 ∪ 𝑋 ∈ SAlg)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝒫 ∪ 𝑋 ∈ SAlg)
14		unipw 5405	. . . . . . 7 ⊢ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋
15	14	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋)
16		pwuni 4905	. . . . . . 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 4878	. . . . . . . 8 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → ∪ 𝑠 = ∪ 𝒫 ∪ 𝑋)
20	19	eqeq1d 2731	. . . . . . 7 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋))
21		sseq2 3970	. . . . . . 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 3656	. . . . 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 4301	. . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
26		intex 5294	. . 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 6957	1 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3402  Vcvv 3444   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  ∪ cuni 4867  ∩ cint 4906   ↦ cmpt 5183  ‘cfv 6499  SAlgcsalg 46299  SalGencsalgen 46303

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 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-salg 46300  df-salgen 46304

This theorem is referenced by:  salgencl  46323  sssalgen  46326  salgenss  46327  salgenuni  46328  issalgend  46329