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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldlfcntref | Structured version Visualization version GIF version |
Description: Every open cover of a LindelΓΆf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.) |
Ref | Expression |
---|---|
ldlfcntref.x | β’ π = βͺ π½ |
Ref | Expression |
---|---|
ldlfcntref | β’ ((π½ β Ldlf β§ π β π½ β§ π = βͺ π) β βπ£ β π« π½(π£ βΌ Ο β§ π£Refπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldlfcntref.x | . 2 β’ π = βͺ π½ | |
2 | df-ldlf 33511 | . 2 β’ Ldlf = CovHasRef{π₯ β£ π₯ βΌ Ο} | |
3 | vex 3467 | . . . 4 β’ π£ β V | |
4 | breq1 5146 | . . . 4 β’ (π₯ = π£ β (π₯ βΌ Ο β π£ βΌ Ο)) | |
5 | 3, 4 | elab 3659 | . . 3 β’ (π£ β {π₯ β£ π₯ βΌ Ο} β π£ βΌ Ο) |
6 | 5 | biimpi 215 | . 2 β’ (π£ β {π₯ β£ π₯ βΌ Ο} β π£ βΌ Ο) |
7 | 1, 2, 6 | crefdf 33506 | 1 β’ ((π½ β Ldlf β§ π β π½ β§ π = βͺ π) β βπ£ β π« π½(π£ βΌ Ο β§ π£Refπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 {cab 2702 βwrex 3060 β wss 3939 π« cpw 4598 βͺ cuni 4903 class class class wbr 5143 Οcom 7868 βΌ cdom 8960 Refcref 23424 Ldlfcldlf 33510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-cref 33501 df-ldlf 33511 |
This theorem is referenced by: (None) |
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