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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 24473 | . . 3 β’ ((π· β (PsMetβπ) β§ π΄ β πΉ) β π΄ β (π Γ π)) |
3 | xpss 5694 | . . 3 β’ (π Γ π) β (V Γ V) | |
4 | 2, 3 | sstrdi 3992 | . 2 β’ ((π· β (PsMetβπ) β§ π΄ β πΉ) β π΄ β (V Γ V)) |
5 | df-rel 5685 | . 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 1534 β wcel 2099 Vcvv 3471 β wss 3947 β¦ cmpt 5231 Γ cxp 5676 β‘ccnv 5677 ran crn 5679 β cima 5681 Rel wrel 5683 βcfv 6548 (class class class)co 7420 0cc0 11139 β+crp 13007 [,)cico 13359 PsMetcpsmet 21263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8847 df-xr 11283 df-psmet 21271 |
This theorem is referenced by: (None) |
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