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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 44738, 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 | β’ (π β π β π) |
salgenss.g | β’ πΊ = (SalGenβπ) |
salgenss.s | β’ (π β π β SAlg) |
salgenss.i | β’ (π β π β π) |
salgenss.u | β’ (π β βͺ π = βͺ π) |
Ref | Expression |
---|---|
salgenss | β’ (π β πΊ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salgenss.g | . . . 4 β’ πΊ = (SalGenβπ) | |
2 | 1 | a1i 11 | . . 3 β’ (π β πΊ = (SalGenβπ)) |
3 | salgenss.x | . . . 4 β’ (π β π β π) | |
4 | salgenval 44715 | . . . 4 β’ (π β π β (SalGenβπ) = β© {π β SAlg β£ (βͺ π = βͺ π β§ π β π )}) | |
5 | 3, 4 | syl 17 | . . 3 β’ (π β (SalGenβπ) = β© {π β SAlg β£ (βͺ π = βͺ π β§ π β π )}) |
6 | 2, 5 | eqtrd 2771 | . 2 β’ (π β πΊ = β© {π β SAlg β£ (βͺ π = βͺ π β§ π β π )}) |
7 | salgenss.s | . . . . 5 β’ (π β π β SAlg) | |
8 | salgenss.u | . . . . . 6 β’ (π β βͺ π = βͺ π) | |
9 | salgenss.i | . . . . . 6 β’ (π β π β π) | |
10 | 8, 9 | jca 512 | . . . . 5 β’ (π β (βͺ π = βͺ π β§ π β π)) |
11 | 7, 10 | jca 512 | . . . 4 β’ (π β (π β SAlg β§ (βͺ π = βͺ π β§ π β π))) |
12 | unieq 4896 | . . . . . . 7 β’ (π = π β βͺ π = βͺ π) | |
13 | 12 | eqeq1d 2733 | . . . . . 6 β’ (π = π β (βͺ π = βͺ π β βͺ π = βͺ π)) |
14 | sseq2 3988 | . . . . . 6 β’ (π = π β (π β π β π β π)) | |
15 | 13, 14 | anbi12d 631 | . . . . 5 β’ (π = π β ((βͺ π = βͺ π β§ π β π ) β (βͺ π = βͺ π β§ π β π))) |
16 | 15 | elrab 3663 | . . . 4 β’ (π β {π β SAlg β£ (βͺ π = βͺ π β§ π β π )} β (π β SAlg β§ (βͺ π = βͺ π β§ π β π))) |
17 | 11, 16 | sylibr 233 | . . 3 β’ (π β π β {π β SAlg β£ (βͺ π = βͺ π β§ π β π )}) |
18 | intss1 4944 | . . 3 β’ (π β {π β SAlg β£ (βͺ π = βͺ π β§ π β π )} β β© {π β SAlg β£ (βͺ π = βͺ π β§ π β π )} β π) | |
19 | 17, 18 | syl 17 | . 2 β’ (π β β© {π β SAlg β£ (βͺ π = βͺ π β§ π β π )} β π) |
20 | 6, 19 | eqsstrd 4000 | 1 β’ (π β πΊ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 {crab 3418 β wss 3928 βͺ cuni 4885 β© cint 4927 βcfv 6516 SAlgcsalg 44702 SalGencsalgen 44706 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-iota 6468 df-fun 6518 df-fv 6524 df-salg 44703 df-salgen 44707 |
This theorem is referenced by: issalgend 44732 dfsalgen2 44735 borelmbl 45030 smfpimbor1lem2 45193 |
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