Theorem salgenss 43765

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 43773, 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 43752	. . . 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 511	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))
11	7, 10	jca 511	. . . 4 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)))
12		unieq 4847	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
13	12	eqeq1d 2740	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑆 = ∪ 𝑋))
14		sseq2 3943	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑆))
15	13, 14	anbi12d 630	. . . . 5 ⊢ (𝑠 = 𝑆 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)))
16	15	elrab 3617	. . . 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 4891	. . 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 3955	1 ⊢ (𝜑 → 𝐺 ⊆ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067   ⊆ wss 3883  ∪ cuni 4836  ∩ cint 4876  ‘cfv 6418  SAlgcsalg 43739  SalGencsalgen 43743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-salg 43740  df-salgen 43744

This theorem is referenced by:  issalgend  43767  dfsalgen2  43770  borelmbl  44064  smfpimbor1lem2  44220