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Mirrors > Home > MPE Home > Th. List > blelrnps | Structured version Visualization version GIF version |
Description: A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
Ref | Expression |
---|---|
blelrnps | β’ ((π· β (PsMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π ) β ran (ballβπ·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blfps 24340 | . . 3 β’ (π· β (PsMetβπ) β (ballβπ·):(π Γ β*)βΆπ« π) | |
2 | 1 | ffnd 6728 | . 2 β’ (π· β (PsMetβπ) β (ballβπ·) Fn (π Γ β*)) |
3 | fnovrn 7603 | . 2 β’ (((ballβπ·) Fn (π Γ β*) β§ π β π β§ π β β*) β (π(ballβπ·)π ) β ran (ballβπ·)) | |
4 | 2, 3 | syl3an1 1160 | 1 β’ ((π· β (PsMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π ) β ran (ballβπ·)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 β wcel 2098 π« cpw 4606 Γ cxp 5680 ran crn 5683 Fn wfn 6548 βcfv 6553 (class class class)co 7426 β*cxr 11287 PsMetcpsmet 21277 ballcbl 21280 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-map 8855 df-xr 11292 df-psmet 21285 df-bl 21288 |
This theorem is referenced by: unirnblps 24353 blssexps 24360 |
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