Theorem salgenss 43417

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 43425, 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 43404	. . . 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 2773	. 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 515	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))
11	7, 10	jca 515	. . . 4 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)))
12		unieq 4807	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
13	12	eqeq1d 2740	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑆 = ∪ 𝑋))
14		sseq2 3903	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑆))
15	13, 14	anbi12d 634	. . . . 5 ⊢ (𝑠 = 𝑆 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)))
16	15	elrab 3588	. . . 4 ⊢ (𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)))
17	11, 16	sylibr 237	. . 3 ⊢ (𝜑 → 𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
18		intss1 4851	. . 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 3915	1 ⊢ (𝜑 → 𝐺 ⊆ 𝑆)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2114  {crab 3057   ⊆ wss 3843  ∪ cuni 4796  ∩ cint 4836  ‘cfv 6339  SAlgcsalg 43391  SalGencsalgen 43395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-int 4837  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-salg 43392  df-salgen 43396

This theorem is referenced by:  issalgend  43419  dfsalgen2  43422  borelmbl  43716  smfpimbor1lem2  43872