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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 14180 | . . . 4 β’ (π· β (PsMetβπ) β (ballβπ·):(π Γ β*)βΆπ« π) | |
| 2 | 1 | frnd 5387 | . . 3 β’ (π· β (PsMetβπ) β ran (ballβπ·) β π« π) |
| 3 | sspwuni 3983 | . . 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 9670 | . . . 4 β’ 1 β β+ | |
| 6 | blcntrps 14186 | . . . 4 β’ ((π· β (PsMetβπ) β§ π₯ β π β§ 1 β β+) β π₯ β (π₯(ballβπ·)1)) | |
| 7 | 5, 6 | mp3an3 1336 | . . 3 β’ ((π· β (PsMetβπ) β§ π₯ β π) β π₯ β (π₯(ballβπ·)1)) |
| 8 | rpxr 9674 | . . . . 5 β’ (1 β β+ β 1 β β*) | |
| 9 | 5, 8 | ax-mp 5 | . . . 4 β’ 1 β β* |
| 10 | blelrnps 14190 | . . . 4 β’ ((π· β (PsMetβπ) β§ π₯ β π β§ 1 β β*) β (π₯(ballβπ·)1) β ran (ballβπ·)) | |
| 11 | 9, 10 | mp3an3 1336 | . . 3 β’ ((π· β (PsMetβπ) β§ π₯ β π) β (π₯(ballβπ·)1) β ran (ballβπ·)) |
| 12 | elunii 3826 | . . 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 3186 | 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 1363 β wcel 2158 β wss 3141 π« cpw 3587 βͺ cuni 3821 Γ cxp 4636 ran crn 4639 βcfv 5228 (class class class)co 5888 1c1 7825 β*cxr 8004 β+crp 9666 PsMetcpsmet 13696 ballcbl 13699 |
| 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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 ax-0lt1 7930 ax-rnegex 7933 |
| This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-map 6663 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-rp 9667 df-psmet 13704 df-bl 13707 |
| This theorem is referenced by: (None) |
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