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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 45358, 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 45335 | . . . 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 2770 | . 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 510 | . . . . 5 β’ (π β (βͺ π = βͺ π β§ π β π)) |
11 | 7, 10 | jca 510 | . . . 4 β’ (π β (π β SAlg β§ (βͺ π = βͺ π β§ π β π))) |
12 | unieq 4918 | . . . . . . 7 β’ (π = π β βͺ π = βͺ π) | |
13 | 12 | eqeq1d 2732 | . . . . . 6 β’ (π = π β (βͺ π = βͺ π β βͺ π = βͺ π)) |
14 | sseq2 4007 | . . . . . 6 β’ (π = π β (π β π β π β π)) | |
15 | 13, 14 | anbi12d 629 | . . . . 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 3682 | . . . 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 4966 | . . 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 4019 | 1 β’ (π β πΊ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 {crab 3430 β wss 3947 βͺ cuni 4907 β© cint 4949 βcfv 6542 SAlgcsalg 45322 SalGencsalgen 45326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-salg 45323 df-salgen 45327 |
This theorem is referenced by: issalgend 45352 dfsalgen2 45355 borelmbl 45650 smfpimbor1lem2 45813 |
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