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Mirrors > Home > MPE Home > Th. List > iscusp | Structured version Visualization version GIF version |
Description: The predicate "π is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.) |
Ref | Expression |
---|---|
iscusp | β’ (π β CUnifSp β (π β UnifSp β§ βπ β (Filβ(Baseβπ))(π β (CauFiluβ(UnifStβπ)) β ((TopOpenβπ) fLim π) β β ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2fveq3 6897 | . . 3 β’ (π€ = π β (Filβ(Baseβπ€)) = (Filβ(Baseβπ))) | |
2 | 2fveq3 6897 | . . . . 5 β’ (π€ = π β (CauFiluβ(UnifStβπ€)) = (CauFiluβ(UnifStβπ))) | |
3 | 2 | eleq2d 2820 | . . . 4 β’ (π€ = π β (π β (CauFiluβ(UnifStβπ€)) β π β (CauFiluβ(UnifStβπ)))) |
4 | fveq2 6892 | . . . . . 6 β’ (π€ = π β (TopOpenβπ€) = (TopOpenβπ)) | |
5 | 4 | oveq1d 7424 | . . . . 5 β’ (π€ = π β ((TopOpenβπ€) fLim π) = ((TopOpenβπ) fLim π)) |
6 | 5 | neeq1d 3001 | . . . 4 β’ (π€ = π β (((TopOpenβπ€) fLim π) β β β ((TopOpenβπ) fLim π) β β )) |
7 | 3, 6 | imbi12d 345 | . . 3 β’ (π€ = π β ((π β (CauFiluβ(UnifStβπ€)) β ((TopOpenβπ€) fLim π) β β ) β (π β (CauFiluβ(UnifStβπ)) β ((TopOpenβπ) fLim π) β β ))) |
8 | 1, 7 | raleqbidv 3343 | . 2 β’ (π€ = π β (βπ β (Filβ(Baseβπ€))(π β (CauFiluβ(UnifStβπ€)) β ((TopOpenβπ€) fLim π) β β ) β βπ β (Filβ(Baseβπ))(π β (CauFiluβ(UnifStβπ)) β ((TopOpenβπ) fLim π) β β ))) |
9 | df-cusp 23803 | . 2 β’ CUnifSp = {π€ β UnifSp β£ βπ β (Filβ(Baseβπ€))(π β (CauFiluβ(UnifStβπ€)) β ((TopOpenβπ€) fLim π) β β )} | |
10 | 8, 9 | elrab2 3687 | 1 β’ (π β CUnifSp β (π β UnifSp β§ βπ β (Filβ(Baseβπ))(π β (CauFiluβ(UnifStβπ)) β ((TopOpenβπ) fLim π) β β ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2941 βwral 3062 β c0 4323 βcfv 6544 (class class class)co 7409 Basecbs 17144 TopOpenctopn 17367 Filcfil 23349 fLim cflim 23438 UnifStcuss 23758 UnifSpcusp 23759 CauFiluccfilu 23791 CUnifSpccusp 23802 |
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-ext 2704 |
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-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 df-cusp 23803 |
This theorem is referenced by: cuspusp 23805 cuspcvg 23806 iscusp2 23807 cmetcusp 24871 |
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