Theorem salgenval 42626

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 42618	. . 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 4849	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ∪ 𝑥 = ∪ 𝑋)
4	3	eqeq2d 2832	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∪ 𝑠 = ∪ 𝑥 ↔ ∪ 𝑠 = ∪ 𝑋))
5		sseq1 3992	. . . . . 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 3480	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)} = {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
8	7	inteqd 4881	. . 3 ⊢ (𝑥 = 𝑋 → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)} = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
9	8	adantl 484	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)} = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
10		elex 3512	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
11		uniexg 7466	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑋 ∈ V)
12		pwsal 42620	. . . . . . 7 ⊢ (∪ 𝑋 ∈ V → 𝒫 ∪ 𝑋 ∈ SAlg)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝒫 ∪ 𝑋 ∈ SAlg)
14		unipw 5343	. . . . . . 7 ⊢ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋
15	14	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋)
16		pwuni 4875	. . . . . . 7 ⊢ 𝑋 ⊆ 𝒫 ∪ 𝑋
17	16	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝒫 ∪ 𝑋)
18	13, 15, 17	jca32 518	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝒫 ∪ 𝑋 ∈ SAlg ∧ (∪ 𝒫 ∪ 𝑋 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝒫 ∪ 𝑋)))
19		unieq 4849	. . . . . . . 8 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → ∪ 𝑠 = ∪ 𝒫 ∪ 𝑋)
20	19	eqeq1d 2823	. . . . . . 7 ⊢ (𝑠 = 𝒫 ∪ 𝑋 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝒫 ∪ 𝑋 = ∪ 𝑋))
21		sseq2 3993	. . . . . . 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 3680	. . . . 5 ⊢ (𝒫 ∪ 𝑋 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝒫 ∪ 𝑋 ∈ SAlg ∧ (∪ 𝒫 ∪ 𝑋 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝒫 ∪ 𝑋)))
24	18, 23	sylibr 236	. . . 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 5240	. . 3 ⊢ ({𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅ ↔ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ∈ V)
27	25, 26	sylib 220	. 2 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ∈ V)
28	2, 9, 10, 27	fvmptd 6775	1 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  {crab 3142  Vcvv 3494   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  ∪ cuni 4838  ∩ cint 4876   ↦ cmpt 5146  ‘cfv 6355  SAlgcsalg 42613  SalGencsalgen 42617

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-int 4877  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-salg 42614  df-salgen 42618

This theorem is referenced by:  salgencl  42635  sssalgen  42638  salgenss  42639  salgenuni  42640  issalgend  42641