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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 7728 | . . . . . 6 β’ βͺ π β V | |
2 | pwsiga 33123 | . . . . . 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 4949 | . . . . 5 β’ π β π« βͺ π | |
5 | sseq2 4008 | . . . . . 6 β’ (π = π« βͺ π β (π β π β π β π« βͺ π)) | |
6 | 5 | rspcev 3612 | . . . . 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 690 | . . . 4 β’ βπ β (sigAlgebraββͺ π)π β π |
8 | rabn0 4385 | . . . 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 5337 | . . 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 33132 | . 2 β’ sigaGen = (π β V β¦ β© {π β (sigAlgebraββͺ π) β£ π β π }) | |
13 | 11, 12 | dmmpti 6694 | 1 β’ dom sigaGen = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 β wcel 2106 β wne 2940 βwrex 3070 {crab 3432 Vcvv 3474 β wss 3948 β c0 4322 π« cpw 4602 βͺ cuni 4908 β© cint 4950 dom cdm 5676 βcfv 6543 sigAlgebracsiga 33101 sigaGencsigagen 33131 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-siga 33102 df-sigagen 33132 |
This theorem is referenced by: (None) |
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