Step | Hyp | Ref
| Expression |
1 | | lfinpfin 22891 |
. . 3
β’ (πΆ β (LocFinβπ«
π) β πΆ β PtFin) |
2 | | unipw 5412 |
. . . . 5
β’ βͺ π« π = π |
3 | 2 | eqcomi 2746 |
. . . 4
β’ π = βͺ
π« π |
4 | | locfindis.1 |
. . . 4
β’ π = βͺ
πΆ |
5 | 3, 4 | locfinbas 22889 |
. . 3
β’ (πΆ β (LocFinβπ«
π) β π = π) |
6 | 1, 5 | jca 513 |
. 2
β’ (πΆ β (LocFinβπ«
π) β (πΆ β PtFin β§ π = π)) |
7 | | simpr 486 |
. . . . 5
β’ ((πΆ β PtFin β§ π = π) β π = π) |
8 | | uniexg 7682 |
. . . . . . 7
β’ (πΆ β PtFin β βͺ πΆ
β V) |
9 | 4, 8 | eqeltrid 2842 |
. . . . . 6
β’ (πΆ β PtFin β π β V) |
10 | 9 | adantr 482 |
. . . . 5
β’ ((πΆ β PtFin β§ π = π) β π β V) |
11 | 7, 10 | eqeltrd 2838 |
. . . 4
β’ ((πΆ β PtFin β§ π = π) β π β V) |
12 | | distop 22361 |
. . . 4
β’ (π β V β π« π β Top) |
13 | 11, 12 | syl 17 |
. . 3
β’ ((πΆ β PtFin β§ π = π) β π« π β Top) |
14 | | snelpwi 5405 |
. . . . . 6
β’ (π₯ β π β {π₯} β π« π) |
15 | 14 | adantl 483 |
. . . . 5
β’ (((πΆ β PtFin β§ π = π) β§ π₯ β π) β {π₯} β π« π) |
16 | | snidg 4625 |
. . . . . 6
β’ (π₯ β π β π₯ β {π₯}) |
17 | 16 | adantl 483 |
. . . . 5
β’ (((πΆ β PtFin β§ π = π) β§ π₯ β π) β π₯ β {π₯}) |
18 | | simpll 766 |
. . . . . 6
β’ (((πΆ β PtFin β§ π = π) β§ π₯ β π) β πΆ β PtFin) |
19 | 7 | eleq2d 2824 |
. . . . . . 7
β’ ((πΆ β PtFin β§ π = π) β (π₯ β π β π₯ β π)) |
20 | 19 | biimpa 478 |
. . . . . 6
β’ (((πΆ β PtFin β§ π = π) β§ π₯ β π) β π₯ β π) |
21 | 4 | ptfinfin 22886 |
. . . . . 6
β’ ((πΆ β PtFin β§ π₯ β π) β {π β πΆ β£ π₯ β π } β Fin) |
22 | 18, 20, 21 | syl2anc 585 |
. . . . 5
β’ (((πΆ β PtFin β§ π = π) β§ π₯ β π) β {π β πΆ β£ π₯ β π } β Fin) |
23 | | eleq2 2827 |
. . . . . . 7
β’ (π¦ = {π₯} β (π₯ β π¦ β π₯ β {π₯})) |
24 | | ineq2 4171 |
. . . . . . . . . . 11
β’ (π¦ = {π₯} β (π β© π¦) = (π β© {π₯})) |
25 | 24 | neeq1d 3004 |
. . . . . . . . . 10
β’ (π¦ = {π₯} β ((π β© π¦) β β
β (π β© {π₯}) β β
)) |
26 | | disjsn 4677 |
. . . . . . . . . . 11
β’ ((π β© {π₯}) = β
β Β¬ π₯ β π ) |
27 | 26 | necon2abii 2995 |
. . . . . . . . . 10
β’ (π₯ β π β (π β© {π₯}) β β
) |
28 | 25, 27 | bitr4di 289 |
. . . . . . . . 9
β’ (π¦ = {π₯} β ((π β© π¦) β β
β π₯ β π )) |
29 | 28 | rabbidv 3418 |
. . . . . . . 8
β’ (π¦ = {π₯} β {π β πΆ β£ (π β© π¦) β β
} = {π β πΆ β£ π₯ β π }) |
30 | 29 | eleq1d 2823 |
. . . . . . 7
β’ (π¦ = {π₯} β ({π β πΆ β£ (π β© π¦) β β
} β Fin β {π β πΆ β£ π₯ β π } β Fin)) |
31 | 23, 30 | anbi12d 632 |
. . . . . 6
β’ (π¦ = {π₯} β ((π₯ β π¦ β§ {π β πΆ β£ (π β© π¦) β β
} β Fin) β (π₯ β {π₯} β§ {π β πΆ β£ π₯ β π } β Fin))) |
32 | 31 | rspcev 3584 |
. . . . 5
β’ (({π₯} β π« π β§ (π₯ β {π₯} β§ {π β πΆ β£ π₯ β π } β Fin)) β βπ¦ β π« π(π₯ β π¦ β§ {π β πΆ β£ (π β© π¦) β β
} β Fin)) |
33 | 15, 17, 22, 32 | syl12anc 836 |
. . . 4
β’ (((πΆ β PtFin β§ π = π) β§ π₯ β π) β βπ¦ β π« π(π₯ β π¦ β§ {π β πΆ β£ (π β© π¦) β β
} β Fin)) |
34 | 33 | ralrimiva 3144 |
. . 3
β’ ((πΆ β PtFin β§ π = π) β βπ₯ β π βπ¦ β π« π(π₯ β π¦ β§ {π β πΆ β£ (π β© π¦) β β
} β Fin)) |
35 | 3, 4 | islocfin 22884 |
. . 3
β’ (πΆ β (LocFinβπ«
π) β (π« π β Top β§ π = π β§ βπ₯ β π βπ¦ β π« π(π₯ β π¦ β§ {π β πΆ β£ (π β© π¦) β β
} β
Fin))) |
36 | 13, 7, 34, 35 | syl3anbrc 1344 |
. 2
β’ ((πΆ β PtFin β§ π = π) β πΆ β (LocFinβπ« π)) |
37 | 6, 36 | impbii 208 |
1
β’ (πΆ β (LocFinβπ«
π) β (πΆ β PtFin β§ π = π)) |