Theorem salgenss 43875

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 43883, 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 43862	. . . 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 2778	. 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 4850	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
13	12	eqeq1d 2740	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑆 = ∪ 𝑋))
14		sseq2 3947	. . . . . 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 3624	. . . 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 4894	. . 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 3959	1 ⊢ (𝜑 → 𝐺 ⊆ 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068   ⊆ wss 3887  ∪ cuni 4839  ∩ cint 4879  ‘cfv 6433  SAlgcsalg 43849  SalGencsalgen 43853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-salg 43850  df-salgen 43854

This theorem is referenced by:  issalgend  43877  dfsalgen2  43880  borelmbl  44174  smfpimbor1lem2  44333