![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 24267 | . . 3 β’ (π· β (PsMetβπ) β (ballβπ·):(π Γ β*)βΆπ« π) | |
2 | 1 | ffnd 6712 | . 2 β’ (π· β (PsMetβπ) β (ballβπ·) Fn (π Γ β*)) |
3 | fnovrn 7579 | . 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 4597 Γ cxp 5667 ran crn 5670 Fn wfn 6532 βcfv 6537 (class class class)co 7405 β*cxr 11251 PsMetcpsmet 21224 ballcbl 21227 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-map 8824 df-xr 11256 df-psmet 21232 df-bl 21235 |
This theorem is referenced by: unirnblps 24280 blssexps 24287 |
Copyright terms: Public domain | W3C validator |