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 39238, 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 39217 | . . . 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 2644 | . 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 553 | . . . . 5 ⊢ (𝜑 → (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)) |
11 | 7, 10 | jca 553 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))) |
12 | unieq 4380 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆) | |
13 | 12 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑆 = ∪ 𝑋)) |
14 | sseq2 3590 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑆)) | |
15 | 13, 14 | anbi12d 743 | . . . . 5 ⊢ (𝑠 = 𝑆 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))) |
16 | 15 | elrab 3331 | . . . 4 ⊢ (𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))) |
17 | 11, 16 | sylibr 223 | . . 3 ⊢ (𝜑 → 𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) |
18 | intss1 4427 | . . 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 3602 | 1 ⊢ (𝜑 → 𝐺 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ⊆ wss 3540 ∪ cuni 4372 ∩ cint 4410 ‘cfv 5804 SAlgcsalg 39204 SalGencsalgen 39208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-salg 39205 df-salgen 39209 |
This theorem is referenced by: issalgend 39232 dfsalgen2 39235 borelmbl 39526 smfpimbor1lem2 39684 |
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