Theorem salgencl 46897

Description: SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Assertion

Ref	Expression
salgencl	⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg)

Proof of Theorem salgencl

Dummy variables 𝑡 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		salgenval 46886	. 2 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
2		ssrab2 4034	. . . 4 ⊢ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ SAlg
3	2	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ SAlg)
4		salgenn0 46896	. . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
5		unieq 4877	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → ∪ 𝑠 = ∪ 𝑡)
6	5	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑡 = ∪ 𝑋))
7		sseq2 3963	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑡))
8	6, 7	anbi12d 641	. . . . . . 7 ⊢ (𝑠 = 𝑡 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
9	8	elrab 3651	. . . . . 6 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑡 ∈ SAlg ∧ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
10	9	biimpi 218	. . . . 5 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → (𝑡 ∈ SAlg ∧ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
11	10	simprld 781	. . . 4 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∪ 𝑡 = ∪ 𝑋)
12	11	adantl 485	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → ∪ 𝑡 = ∪ 𝑋)
13	3, 4, 12	intsal 46895	. 2 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ∈ SAlg)
14	1, 13	eqeltrd 2863	1 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  {crab 3415   ⊆ wss 3905  ∪ cuni 4866  ∩ cint 4906  ‘cfv 6521  SAlgcsalg 46873  SalGencsalgen 46877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-iota 6477  df-fun 6523  df-fv 6529  df-salg 46874  df-salgen 46878

This theorem is referenced by:  unisalgen  46905  dfsalgen2  46906  salgencld  46914