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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigagenss | Structured version Visualization version GIF version |
Description: The generated sigma-algebra is a subset of all sigma-algebras containing the generating set, i.e. the generated sigma-algebra is the smallest sigma-algebra containing the generating set, here π΄. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
Ref | Expression |
---|---|
sigagenss | β’ ((π β (sigAlgebraββͺ π΄) β§ π΄ β π) β (sigaGenβπ΄) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 5284 | . . . 4 β’ ((π΄ β π β§ π β (sigAlgebraββͺ π΄)) β π΄ β V) | |
2 | 1 | ancoms 460 | . . 3 β’ ((π β (sigAlgebraββͺ π΄) β§ π΄ β π) β π΄ β V) |
3 | sigagenval 32803 | . . 3 β’ (π΄ β V β (sigaGenβπ΄) = β© {π β (sigAlgebraββͺ π΄) β£ π΄ β π }) | |
4 | 2, 3 | syl 17 | . 2 β’ ((π β (sigAlgebraββͺ π΄) β§ π΄ β π) β (sigaGenβπ΄) = β© {π β (sigAlgebraββͺ π΄) β£ π΄ β π }) |
5 | sseq2 3974 | . . 3 β’ (π = π β (π΄ β π β π΄ β π)) | |
6 | 5 | intminss 4939 | . 2 β’ ((π β (sigAlgebraββͺ π΄) β§ π΄ β π) β β© {π β (sigAlgebraββͺ π΄) β£ π΄ β π } β π) |
7 | 4, 6 | eqsstrd 3986 | 1 β’ ((π β (sigAlgebraββͺ π΄) β§ π΄ β π) β (sigaGenβπ΄) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3406 Vcvv 3447 β wss 3914 βͺ cuni 4869 β© cint 4911 βcfv 6500 sigAlgebracsiga 32771 sigaGencsigagen 32801 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-siga 32772 df-sigagen 32802 |
This theorem is referenced by: sigagenss2 32813 sigagenid 32814 imambfm 32926 |
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