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Mirrors > Home > MPE Home > Th. List > metustrel | Structured version Visualization version GIF version |
Description: Elements of the filter base generated by the metric π· are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
Ref | Expression |
---|---|
metust.1 | β’ πΉ = ran (π β β+ β¦ (β‘π· β (0[,)π))) |
Ref | Expression |
---|---|
metustrel | β’ ((π· β (PsMetβπ) β§ π΄ β πΉ) β Rel π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metust.1 | . . . 4 β’ πΉ = ran (π β β+ β¦ (β‘π· β (0[,)π))) | |
2 | 1 | metustss 24411 | . . 3 β’ ((π· β (PsMetβπ) β§ π΄ β πΉ) β π΄ β (π Γ π)) |
3 | xpss 5685 | . . 3 β’ (π Γ π) β (V Γ V) | |
4 | 2, 3 | sstrdi 3989 | . 2 β’ ((π· β (PsMetβπ) β§ π΄ β πΉ) β π΄ β (V Γ V)) |
5 | df-rel 5676 | . 2 β’ (Rel π΄ β π΄ β (V Γ V)) | |
6 | 4, 5 | sylibr 233 | 1 β’ ((π· β (PsMetβπ) β§ π΄ β πΉ) β Rel π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 β¦ cmpt 5224 Γ cxp 5667 β‘ccnv 5668 ran crn 5670 β cima 5672 Rel wrel 5674 βcfv 6536 (class class class)co 7404 0cc0 11109 β+crp 12977 [,)cico 13329 PsMetcpsmet 21220 |
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 7721 ax-cnex 11165 ax-resscn 11166 |
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-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-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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 df-xr 11253 df-psmet 21228 |
This theorem is referenced by: (None) |
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