Theorem salgenss 44825

Description: The sigma-algebra generated by a set is the smallest sigma-algebra, on the same base set, that includes the set. Proposition 111G (b) of [Fremlin1] p. 13. Notice that the condition "on the same base set" is needed, see the counterexample salgensscntex 44833, where a sigma-algebra is shown that includes a set, but does not include the sigma-algebra generated (the key is that its base set is larger than the base set of the generating set). (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
salgenss.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
salgenss.g	⊢ 𝐺 = (SalGen‘𝑋)
salgenss.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
salgenss.i	⊢ (𝜑 → 𝑋 ⊆ 𝑆)
salgenss.u	⊢ (𝜑 → ∪ 𝑆 = ∪ 𝑋)

Assertion

Ref	Expression
salgenss	⊢ (𝜑 → 𝐺 ⊆ 𝑆)

Proof of Theorem salgenss

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		salgenss.g	. . . 4 ⊢ 𝐺 = (SalGen‘𝑋)
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐺 = (SalGen‘𝑋))
3		salgenss.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
4		salgenval 44810	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
6	2, 5	eqtrd 2771	. 2 ⊢ (𝜑 → 𝐺 = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
7		salgenss.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg)
8		salgenss.u	. . . . . 6 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝑋)
9		salgenss.i	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
10	8, 9	jca 512	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))
11	7, 10	jca 512	. . . 4 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)))
12		unieq 4912	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
13	12	eqeq1d 2733	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑆 = ∪ 𝑋))
14		sseq2 4004	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑆))
15	13, 14	anbi12d 631	. . . . 5 ⊢ (𝑠 = 𝑆 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)))
16	15	elrab 3679	. . . 4 ⊢ (𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)))
17	11, 16	sylibr 233	. . 3 ⊢ (𝜑 → 𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
18		intss1 4960	. . 3 ⊢ (𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ 𝑆)
19	17, 18	syl 17	. 2 ⊢ (𝜑 → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ 𝑆)
20	6, 19	eqsstrd 4016	1 ⊢ (𝜑 → 𝐺 ⊆ 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3431   ⊆ wss 3944  ∪ cuni 4901  ∩ cint 4943  ‘cfv 6532  SAlgcsalg 44797  SalGencsalgen 44801

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-salg 44798  df-salgen 44802

This theorem is referenced by:  issalgend  44827  dfsalgen2  44830  borelmbl  45125  smfpimbor1lem2  45288