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| Mirrors > Home > ILE Home > Th. List > unirnblps | 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.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
| Ref | Expression |
|---|---|
| unirnblps | β’ (π· β (PsMetβπ) β βͺ ran (ballβπ·) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | blfps 14312 | . . . 4 β’ (π· β (PsMetβπ) β (ballβπ·):(π Γ β*)βΆπ« π) | |
| 2 | 1 | frnd 5390 | . . 3 β’ (π· β (PsMetβπ) β ran (ballβπ·) β π« π) |
| 3 | sspwuni 3986 | . . 3 β’ (ran (ballβπ·) β π« π β βͺ ran (ballβπ·) β π) | |
| 4 | 2, 3 | sylib 122 | . 2 β’ (π· β (PsMetβπ) β βͺ ran (ballβπ·) β π) |
| 5 | 1rp 9676 | . . . 4 β’ 1 β β+ | |
| 6 | blcntrps 14318 | . . . 4 β’ ((π· β (PsMetβπ) β§ π₯ β π β§ 1 β β+) β π₯ β (π₯(ballβπ·)1)) | |
| 7 | 5, 6 | mp3an3 1337 | . . 3 β’ ((π· β (PsMetβπ) β§ π₯ β π) β π₯ β (π₯(ballβπ·)1)) |
| 8 | rpxr 9680 | . . . . 5 β’ (1 β β+ β 1 β β*) | |
| 9 | 5, 8 | ax-mp 5 | . . . 4 β’ 1 β β* |
| 10 | blelrnps 14322 | . . . 4 β’ ((π· β (PsMetβπ) β§ π₯ β π β§ 1 β β*) β (π₯(ballβπ·)1) β ran (ballβπ·)) | |
| 11 | 9, 10 | mp3an3 1337 | . . 3 β’ ((π· β (PsMetβπ) β§ π₯ β π) β (π₯(ballβπ·)1) β ran (ballβπ·)) |
| 12 | elunii 3829 | . . 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 (PsMetβπ) β§ π₯ β π) β π₯ β βͺ ran (ballβπ·)) |
| 14 | 4, 13 | eqelssd 3189 | 1 β’ (π· β (PsMetβπ) β βͺ ran (ballβπ·) = π) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 = wceq 1364 β wcel 2160 β wss 3144 π« cpw 3590 βͺ cuni 3824 Γ cxp 4639 ran crn 4642 βcfv 5231 (class class class)co 5891 1c1 7831 β*cxr 8010 β+crp 9672 PsMetcpsmet 13815 ballcbl 13818 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1re 7924 ax-addrcl 7927 ax-0lt1 7936 ax-rnegex 7939 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-map 6668 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-rp 9673 df-psmet 13823 df-bl 13826 |
| This theorem is referenced by: (None) |
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