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| Mirrors > Home > MPE Home > Th. List > unirnblps | Structured version Visualization version 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 24233 | . . . 4 β’ (π· β (PsMetβπ) β (ballβπ·):(π Γ β*)βΆπ« π) | |
| 2 | 1 | frnd 6715 | . . 3 β’ (π· β (PsMetβπ) β ran (ballβπ·) β π« π) |
| 3 | sspwuni 5093 | . . 3 β’ (ran (ballβπ·) β π« π β βͺ ran (ballβπ·) β π) | |
| 4 | 2, 3 | sylib 217 | . 2 β’ (π· β (PsMetβπ) β βͺ ran (ballβπ·) β π) |
| 5 | 1rp 12974 | . . . 4 β’ 1 β β+ | |
| 6 | blcntrps 24239 | . . . 4 β’ ((π· β (PsMetβπ) β§ π₯ β π β§ 1 β β+) β π₯ β (π₯(ballβπ·)1)) | |
| 7 | 5, 6 | mp3an3 1446 | . . 3 β’ ((π· β (PsMetβπ) β§ π₯ β π) β π₯ β (π₯(ballβπ·)1)) |
| 8 | 1xr 11269 | . . . 4 β’ 1 β β* | |
| 9 | blelrnps 24243 | . . . 4 β’ ((π· β (PsMetβπ) β§ π₯ β π β§ 1 β β*) β (π₯(ballβπ·)1) β ran (ballβπ·)) | |
| 10 | 8, 9 | mp3an3 1446 | . . 3 β’ ((π· β (PsMetβπ) β§ π₯ β π) β (π₯(ballβπ·)1) β ran (ballβπ·)) |
| 11 | elunii 4904 | . . 3 β’ ((π₯ β (π₯(ballβπ·)1) β§ (π₯(ballβπ·)1) β ran (ballβπ·)) β π₯ β βͺ ran (ballβπ·)) | |
| 12 | 7, 10, 11 | syl2anc 583 | . 2 β’ ((π· β (PsMetβπ) β§ π₯ β π) β π₯ β βͺ ran (ballβπ·)) |
| 13 | 4, 12 | eqelssd 3995 | 1 β’ (π· β (PsMetβπ) β βͺ ran (ballβπ·) = π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3940 π« cpw 4594 βͺ cuni 4899 Γ cxp 5664 ran crn 5667 βcfv 6533 (class class class)co 7401 1c1 11106 β*cxr 11243 β+crp 12970 PsMetcpsmet 21211 ballcbl 21214 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-rp 12971 df-psmet 21219 df-bl 21222 |
| This theorem is referenced by: psmetutop 24397 |
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