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Mirrors > Home > ILE Home > Th. List > blrnps | GIF version |
Description: Membership in the range of the ball function. Note that ran (ballβπ·) is the collection of all balls for metric π·. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
Ref | Expression |
---|---|
blrnps | β’ (π· β (PsMetβπ) β (π΄ β ran (ballβπ·) β βπ₯ β π βπ β β* π΄ = (π₯(ballβπ·)π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blfps 14293 | . 2 β’ (π· β (PsMetβπ) β (ballβπ·):(π Γ β*)βΆπ« π) | |
2 | ffn 5380 | . 2 β’ ((ballβπ·):(π Γ β*)βΆπ« π β (ballβπ·) Fn (π Γ β*)) | |
3 | ovelrn 6040 | . 2 β’ ((ballβπ·) Fn (π Γ β*) β (π΄ β ran (ballβπ·) β βπ₯ β π βπ β β* π΄ = (π₯(ballβπ·)π))) | |
4 | 1, 2, 3 | 3syl 17 | 1 β’ (π· β (PsMetβπ) β (π΄ β ran (ballβπ·) β βπ₯ β π βπ β β* π΄ = (π₯(ballβπ·)π))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β wb 105 = wceq 1364 β wcel 2160 βwrex 2469 π« cpw 3590 Γ cxp 4639 ran crn 4642 Fn wfn 5226 βΆwf 5227 βcfv 5231 (class class class)co 5891 β*cxr 8009 PsMetcpsmet 13809 ballcbl 13812 |
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 7920 ax-resscn 7921 |
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-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 8012 df-mnf 8013 df-xr 8014 df-psmet 13817 df-bl 13820 |
This theorem is referenced by: blssps 14311 |
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