Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salgenss | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
salgenss | ⊢ (𝜑 → 𝐺 ⊆ 𝑆) |
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) |
Colors of variables: wff setvar class |
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 |
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