Description: The locally finite covers of a discrete space are precisely the point-finite covers. (Contributed by Jeff Hankins, 22-Jan-2010) (Proof shortened by Mario Carneiro, 11-Sep-2015)
Ref | Expression | ||
---|---|---|---|
Hypothesis | locfindis.1 | |
|
Assertion | locfindis | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | locfindis.1 | |
|
2 | lfinpfin | |
|
3 | unipw | |
|
4 | 3 | eqcomi | |
5 | 4 1 | locfinbas | |
6 | 2 5 | jca | |
7 | simpr | |
|
8 | uniexg | |
|
9 | 1 8 | eqeltrid | |
10 | 9 | adantr | |
11 | 7 10 | eqeltrd | |
12 | distop | |
|
13 | 11 12 | syl | |
14 | snelpwi | |
|
15 | 14 | adantl | |
16 | snidg | |
|
17 | 16 | adantl | |
18 | simpll | |
|
19 | 7 | eleq2d | |
20 | 19 | biimpa | |
21 | 1 | ptfinfin | |
22 | 18 20 21 | syl2anc | |
23 | eleq2 | |
|
24 | ineq2 | |
|
25 | 24 | neeq1d | |
26 | disjsn | |
|
27 | 26 | necon2abii | |
28 | 25 27 | bitr4di | |
29 | 28 | rabbidv | |
30 | 29 | eleq1d | |
31 | 23 30 | anbi12d | |
32 | 31 | rspcev | |
33 | 15 17 22 32 | syl12anc | |
34 | 33 | ralrimiva | |
35 | 4 1 | islocfin | |
36 | 13 7 34 35 | syl3anbrc | |
37 | 6 36 | impbii | |