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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 6896 | . . 3 β’ (π€ = π β (Filβ(Baseβπ€)) = (Filβ(Baseβπ))) | |
2 | 2fveq3 6896 | . . . . 5 β’ (π€ = π β (CauFiluβ(UnifStβπ€)) = (CauFiluβ(UnifStβπ))) | |
3 | 2 | eleq2d 2819 | . . . 4 β’ (π€ = π β (π β (CauFiluβ(UnifStβπ€)) β π β (CauFiluβ(UnifStβπ)))) |
4 | fveq2 6891 | . . . . . 6 β’ (π€ = π β (TopOpenβπ€) = (TopOpenβπ)) | |
5 | 4 | oveq1d 7426 | . . . . 5 β’ (π€ = π β ((TopOpenβπ€) fLim π) = ((TopOpenβπ) fLim π)) |
6 | 5 | neeq1d 3000 | . . . 4 β’ (π€ = π β (((TopOpenβπ€) fLim π) β β β ((TopOpenβπ) fLim π) β β )) |
7 | 3, 6 | imbi12d 344 | . . 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 3342 | . 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 24023 | . 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 3686 | 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 396 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 β c0 4322 βcfv 6543 (class class class)co 7411 Basecbs 17148 TopOpenctopn 17371 Filcfil 23569 fLim cflim 23658 UnifStcuss 23978 UnifSpcusp 23979 CauFiluccfilu 24011 CUnifSpccusp 24022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7414 df-cusp 24023 |
This theorem is referenced by: cuspusp 24025 cuspcvg 24026 iscusp2 24027 cmetcusp 25095 |
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