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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ispcmp | Structured version Visualization version GIF version |
Description: The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.) |
Ref | Expression |
---|---|
ispcmp | β’ (π½ β Paracomp β π½ β CovHasRef(LocFinβπ½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3461 | . 2 β’ (π½ β Paracomp β π½ β V) | |
2 | elex 3461 | . 2 β’ (π½ β CovHasRef(LocFinβπ½) β π½ β V) | |
3 | id 22 | . . . 4 β’ (π = π½ β π = π½) | |
4 | fveq2 6839 | . . . . 5 β’ (π = π½ β (LocFinβπ) = (LocFinβπ½)) | |
5 | crefeq 32238 | . . . . 5 β’ ((LocFinβπ) = (LocFinβπ½) β CovHasRef(LocFinβπ) = CovHasRef(LocFinβπ½)) | |
6 | 4, 5 | syl 17 | . . . 4 β’ (π = π½ β CovHasRef(LocFinβπ) = CovHasRef(LocFinβπ½)) |
7 | 3, 6 | eleq12d 2832 | . . 3 β’ (π = π½ β (π β CovHasRef(LocFinβπ) β π½ β CovHasRef(LocFinβπ½))) |
8 | df-pcmp 32249 | . . 3 β’ Paracomp = {π β£ π β CovHasRef(LocFinβπ)} | |
9 | 7, 8 | elab2g 3630 | . 2 β’ (π½ β V β (π½ β Paracomp β π½ β CovHasRef(LocFinβπ½))) |
10 | 1, 2, 9 | pm5.21nii 379 | 1 β’ (π½ β Paracomp β π½ β CovHasRef(LocFinβπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1541 β wcel 2106 Vcvv 3443 βcfv 6493 LocFinclocfin 22807 CovHasRefccref 32235 Paracompcpcmp 32248 |
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 2708 |
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 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-cref 32236 df-pcmp 32249 |
This theorem is referenced by: cmppcmp 32251 dispcmp 32252 pcmplfin 32253 |
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