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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmsigagen | Structured version Visualization version GIF version |
Description: A sigma-algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
Ref | Expression |
---|---|
dmsigagen | β’ dom sigaGen = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vuniex 7680 | . . . . . 6 β’ βͺ π β V | |
2 | pwsiga 32793 | . . . . . 6 β’ (βͺ π β V β π« βͺ π β (sigAlgebraββͺ π)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 β’ π« βͺ π β (sigAlgebraββͺ π) |
4 | pwuni 4910 | . . . . 5 β’ π β π« βͺ π | |
5 | sseq2 3974 | . . . . . 6 β’ (π = π« βͺ π β (π β π β π β π« βͺ π)) | |
6 | 5 | rspcev 3583 | . . . . 5 β’ ((π« βͺ π β (sigAlgebraββͺ π) β§ π β π« βͺ π) β βπ β (sigAlgebraββͺ π)π β π ) |
7 | 3, 4, 6 | mp2an 691 | . . . 4 β’ βπ β (sigAlgebraββͺ π)π β π |
8 | rabn0 4349 | . . . 4 β’ ({π β (sigAlgebraββͺ π) β£ π β π } β β β βπ β (sigAlgebraββͺ π)π β π ) | |
9 | 7, 8 | mpbir 230 | . . 3 β’ {π β (sigAlgebraββͺ π) β£ π β π } β β |
10 | intex 5298 | . . 3 β’ ({π β (sigAlgebraββͺ π) β£ π β π } β β β β© {π β (sigAlgebraββͺ π) β£ π β π } β V) | |
11 | 9, 10 | mpbi 229 | . 2 β’ β© {π β (sigAlgebraββͺ π) β£ π β π } β V |
12 | df-sigagen 32802 | . 2 β’ sigaGen = (π β V β¦ β© {π β (sigAlgebraββͺ π) β£ π β π }) | |
13 | 11, 12 | dmmpti 6649 | 1 β’ dom sigaGen = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 β wne 2940 βwrex 3070 {crab 3406 Vcvv 3447 β wss 3914 β c0 4286 π« cpw 4564 βͺ cuni 4869 β© cint 4911 dom cdm 5637 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-fn 6503 df-fv 6508 df-siga 32772 df-sigagen 32802 |
This theorem is referenced by: (None) |
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