Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 42634, 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 42613 | . . . 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 2858 | . 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 514 | . . . . 5 ⊢ (𝜑 → (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)) |
11 | 7, 10 | jca 514 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))) |
12 | unieq 4851 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆) | |
13 | 12 | eqeq1d 2825 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑆 = ∪ 𝑋)) |
14 | sseq2 3995 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑆)) | |
15 | 13, 14 | anbi12d 632 | . . . . 5 ⊢ (𝑠 = 𝑆 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))) |
16 | 15 | elrab 3682 | . . . 4 ⊢ (𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))) |
17 | 11, 16 | sylibr 236 | . . 3 ⊢ (𝜑 → 𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) |
18 | intss1 4893 | . . 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 4007 | 1 ⊢ (𝜑 → 𝐺 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 ⊆ wss 3938 ∪ cuni 4840 ∩ cint 4878 ‘cfv 6357 SAlgcsalg 42600 SalGencsalgen 42604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-salg 42601 df-salgen 42605 |
This theorem is referenced by: issalgend 42628 dfsalgen2 42631 borelmbl 42925 smfpimbor1lem2 43081 |
Copyright terms: Public domain | W3C validator |