Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > unirnbl | GIF version | ||
| Description: The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
| Ref | Expression |
|---|---|
| unirnbl | β’ (π· β (βMetβπ) β βͺ ran (ballβπ·) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | blf 13995 | . . . 4 β’ (π· β (βMetβπ) β (ballβπ·):(π Γ β*)βΆπ« π) | |
| 2 | 1 | frnd 5377 | . . 3 β’ (π· β (βMetβπ) β ran (ballβπ·) β π« π) |
| 3 | sspwuni 3973 | . . 3 β’ (ran (ballβπ·) β π« π β βͺ ran (ballβπ·) β π) | |
| 4 | 2, 3 | sylib 122 | . 2 β’ (π· β (βMetβπ) β βͺ ran (ballβπ·) β π) |
| 5 | 1rp 9659 | . . . 4 β’ 1 β β+ | |
| 6 | blcntr 14001 | . . . 4 β’ ((π· β (βMetβπ) β§ π₯ β π β§ 1 β β+) β π₯ β (π₯(ballβπ·)1)) | |
| 7 | 5, 6 | mp3an3 1326 | . . 3 β’ ((π· β (βMetβπ) β§ π₯ β π) β π₯ β (π₯(ballβπ·)1)) |
| 8 | rpxr 9663 | . . . . 5 β’ (1 β β+ β 1 β β*) | |
| 9 | 5, 8 | ax-mp 5 | . . . 4 β’ 1 β β* |
| 10 | blelrn 14005 | . . . 4 β’ ((π· β (βMetβπ) β§ π₯ β π β§ 1 β β*) β (π₯(ballβπ·)1) β ran (ballβπ·)) | |
| 11 | 9, 10 | mp3an3 1326 | . . 3 β’ ((π· β (βMetβπ) β§ π₯ β π) β (π₯(ballβπ·)1) β ran (ballβπ·)) |
| 12 | elunii 3816 | . . 3 β’ ((π₯ β (π₯(ballβπ·)1) β§ (π₯(ballβπ·)1) β ran (ballβπ·)) β π₯ β βͺ ran (ballβπ·)) | |
| 13 | 7, 11, 12 | syl2anc 411 | . 2 β’ ((π· β (βMetβπ) β§ π₯ β π) β π₯ β βͺ ran (ballβπ·)) |
| 14 | 4, 13 | eqelssd 3176 | 1 β’ (π· β (βMetβπ) β βͺ ran (ballβπ·) = π) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 β wss 3131 π« cpw 3577 βͺ cuni 3811 Γ cxp 4626 ran crn 4629 βcfv 5218 (class class class)co 5877 1c1 7814 β*cxr 7993 β+crp 9655 βMetcxmet 13525 ballcbl 13527 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1re 7907 ax-addrcl 7910 ax-0lt1 7919 ax-rnegex 7922 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-map 6652 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-rp 9656 df-psmet 13532 df-xmet 13533 df-bl 13535 |
| This theorem is referenced by: blbas 14018 mopntopon 14028 elmopn 14031 metss 14079 xmettx 14095 |
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