Theorem salgenss 46764

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 46772, 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 46749	. . . 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 511	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))
11	7, 10	jca 511	. . . 4 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)))
12		unieq 4861	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
13	12	eqeq1d 2738	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑆 = ∪ 𝑋))
14		sseq2 3948	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑆))
15	13, 14	anbi12d 633	. . . . 5 ⊢ (𝑠 = 𝑆 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)))
16	15	elrab 3634	. . . 4 ⊢ (𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)))
17	11, 16	sylibr 234	. . 3 ⊢ (𝜑 → 𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
18		intss1 4905	. . 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 3956	1 ⊢ (𝜑 → 𝐺 ⊆ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3389   ⊆ wss 3889  ∪ cuni 4850  ∩ cint 4889  ‘cfv 6498  SAlgcsalg 46736  SalGencsalgen 46740

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-salg 46737  df-salgen 46741

This theorem is referenced by:  issalgend  46766  dfsalgen2  46769  borelmbl  47064  smfpimbor1lem2  47227