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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 23930 | . . 3 β’ ((π· β (PsMetβπ) β§ π΄ β πΉ) β π΄ β (π Γ π)) |
3 | xpss 5653 | . . 3 β’ (π Γ π) β (V Γ V) | |
4 | 2, 3 | sstrdi 3960 | . 2 β’ ((π· β (PsMetβπ) β§ π΄ β πΉ) β π΄ β (V Γ V)) |
5 | df-rel 5644 | . 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 397 = wceq 1542 β wcel 2107 Vcvv 3447 β wss 3914 β¦ cmpt 5192 Γ cxp 5635 β‘ccnv 5636 ran crn 5638 β cima 5640 Rel wrel 5642 βcfv 6500 (class class class)co 7361 0cc0 11059 β+crp 12923 [,)cico 13275 PsMetcpsmet 20803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-map 8773 df-xr 11201 df-psmet 20811 |
This theorem is referenced by: (None) |
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