Step | Hyp | Ref
| Expression |
1 | | distop 22361 |
. 2
β’ (π β π β π« π β Top) |
2 | | eqidd 2738 |
. 2
β’ (π β π β π = π) |
3 | | snelpwi 5405 |
. . . . 5
β’ (π§ β π β {π§} β π« π) |
4 | 3 | adantl 483 |
. . . 4
β’ ((π β π β§ π§ β π) β {π§} β π« π) |
5 | | vsnid 4628 |
. . . . 5
β’ π§ β {π§} |
6 | 5 | a1i 11 |
. . . 4
β’ ((π β π β§ π§ β π) β π§ β {π§}) |
7 | | nfv 1918 |
. . . . . 6
β’
β²π’(π β π β§ π§ β π) |
8 | | nfrab1 3429 |
. . . . . 6
β’
β²π’{π’ β πΆ β£ (π’ β© {π§}) β β
} |
9 | | nfcv 2908 |
. . . . . 6
β’
β²π’{{π§}} |
10 | | dissnref.c |
. . . . . . . . . 10
β’ πΆ = {π’ β£ βπ₯ β π π’ = {π₯}} |
11 | 10 | eqabi 2882 |
. . . . . . . . 9
β’ (π’ β πΆ β βπ₯ β π π’ = {π₯}) |
12 | 11 | anbi1i 625 |
. . . . . . . 8
β’ ((π’ β πΆ β§ (π’ β© {π§}) β β
) β (βπ₯ β π π’ = {π₯} β§ (π’ β© {π§}) β β
)) |
13 | | simpr 486 |
. . . . . . . . . . . . 13
β’
(((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ π₯ β π) β§ π’ = {π₯}) β π’ = {π₯}) |
14 | | simplr 768 |
. . . . . . . . . . . . . . . . . 18
β’
((((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ π₯ β π) β§ π’ = {π₯}) β§ π₯ β π§) β π’ = {π₯}) |
15 | 14 | ineq1d 4176 |
. . . . . . . . . . . . . . . . 17
β’
((((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ π₯ β π) β§ π’ = {π₯}) β§ π₯ β π§) β (π’ β© {π§}) = ({π₯} β© {π§})) |
16 | | disjsn2 4678 |
. . . . . . . . . . . . . . . . . 18
β’ (π₯ β π§ β ({π₯} β© {π§}) = β
) |
17 | 16 | adantl 483 |
. . . . . . . . . . . . . . . . 17
β’
((((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ π₯ β π) β§ π’ = {π₯}) β§ π₯ β π§) β ({π₯} β© {π§}) = β
) |
18 | 15, 17 | eqtrd 2777 |
. . . . . . . . . . . . . . . 16
β’
((((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ π₯ β π) β§ π’ = {π₯}) β§ π₯ β π§) β (π’ β© {π§}) = β
) |
19 | | simp-4r 783 |
. . . . . . . . . . . . . . . . 17
β’
((((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ π₯ β π) β§ π’ = {π₯}) β§ π₯ β π§) β (π’ β© {π§}) β β
) |
20 | 19 | neneqd 2949 |
. . . . . . . . . . . . . . . 16
β’
((((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ π₯ β π) β§ π’ = {π₯}) β§ π₯ β π§) β Β¬ (π’ β© {π§}) = β
) |
21 | 18, 20 | pm2.65da 816 |
. . . . . . . . . . . . . . 15
β’
(((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ π₯ β π) β§ π’ = {π₯}) β Β¬ π₯ β π§) |
22 | | nne 2948 |
. . . . . . . . . . . . . . 15
β’ (Β¬
π₯ β π§ β π₯ = π§) |
23 | 21, 22 | sylib 217 |
. . . . . . . . . . . . . 14
β’
(((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ π₯ β π) β§ π’ = {π₯}) β π₯ = π§) |
24 | 23 | sneqd 4603 |
. . . . . . . . . . . . 13
β’
(((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ π₯ β π) β§ π’ = {π₯}) β {π₯} = {π§}) |
25 | 13, 24 | eqtrd 2777 |
. . . . . . . . . . . 12
β’
(((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ π₯ β π) β§ π’ = {π₯}) β π’ = {π§}) |
26 | 25 | r19.29an 3156 |
. . . . . . . . . . 11
β’ ((((π β π β§ π§ β π) β§ (π’ β© {π§}) β β
) β§ βπ₯ β π π’ = {π₯}) β π’ = {π§}) |
27 | 26 | an32s 651 |
. . . . . . . . . 10
β’ ((((π β π β§ π§ β π) β§ βπ₯ β π π’ = {π₯}) β§ (π’ β© {π§}) β β
) β π’ = {π§}) |
28 | 27 | anasss 468 |
. . . . . . . . 9
β’ (((π β π β§ π§ β π) β§ (βπ₯ β π π’ = {π₯} β§ (π’ β© {π§}) β β
)) β π’ = {π§}) |
29 | | sneq 4601 |
. . . . . . . . . . . 12
β’ (π₯ = π§ β {π₯} = {π§}) |
30 | 29 | rspceeqv 3600 |
. . . . . . . . . . 11
β’ ((π§ β π β§ π’ = {π§}) β βπ₯ β π π’ = {π₯}) |
31 | 30 | adantll 713 |
. . . . . . . . . 10
β’ (((π β π β§ π§ β π) β§ π’ = {π§}) β βπ₯ β π π’ = {π₯}) |
32 | | simpr 486 |
. . . . . . . . . . . . 13
β’ (((π β π β§ π§ β π) β§ π’ = {π§}) β π’ = {π§}) |
33 | 32 | ineq1d 4176 |
. . . . . . . . . . . 12
β’ (((π β π β§ π§ β π) β§ π’ = {π§}) β (π’ β© {π§}) = ({π§} β© {π§})) |
34 | | inidm 4183 |
. . . . . . . . . . . 12
β’ ({π§} β© {π§}) = {π§} |
35 | 33, 34 | eqtrdi 2793 |
. . . . . . . . . . 11
β’ (((π β π β§ π§ β π) β§ π’ = {π§}) β (π’ β© {π§}) = {π§}) |
36 | | vex 3452 |
. . . . . . . . . . . . 13
β’ π§ β V |
37 | 36 | snnz 4742 |
. . . . . . . . . . . 12
β’ {π§} β β
|
38 | 37 | a1i 11 |
. . . . . . . . . . 11
β’ (((π β π β§ π§ β π) β§ π’ = {π§}) β {π§} β β
) |
39 | 35, 38 | eqnetrd 3012 |
. . . . . . . . . 10
β’ (((π β π β§ π§ β π) β§ π’ = {π§}) β (π’ β© {π§}) β β
) |
40 | 31, 39 | jca 513 |
. . . . . . . . 9
β’ (((π β π β§ π§ β π) β§ π’ = {π§}) β (βπ₯ β π π’ = {π₯} β§ (π’ β© {π§}) β β
)) |
41 | 28, 40 | impbida 800 |
. . . . . . . 8
β’ ((π β π β§ π§ β π) β ((βπ₯ β π π’ = {π₯} β§ (π’ β© {π§}) β β
) β π’ = {π§})) |
42 | 12, 41 | bitrid 283 |
. . . . . . 7
β’ ((π β π β§ π§ β π) β ((π’ β πΆ β§ (π’ β© {π§}) β β
) β π’ = {π§})) |
43 | | rabid 3430 |
. . . . . . 7
β’ (π’ β {π’ β πΆ β£ (π’ β© {π§}) β β
} β (π’ β πΆ β§ (π’ β© {π§}) β β
)) |
44 | | velsn 4607 |
. . . . . . 7
β’ (π’ β {{π§}} β π’ = {π§}) |
45 | 42, 43, 44 | 3bitr4g 314 |
. . . . . 6
β’ ((π β π β§ π§ β π) β (π’ β {π’ β πΆ β£ (π’ β© {π§}) β β
} β π’ β {{π§}})) |
46 | 7, 8, 9, 45 | eqrd 3968 |
. . . . 5
β’ ((π β π β§ π§ β π) β {π’ β πΆ β£ (π’ β© {π§}) β β
} = {{π§}}) |
47 | | snfi 8995 |
. . . . 5
β’ {{π§}} β Fin |
48 | 46, 47 | eqeltrdi 2846 |
. . . 4
β’ ((π β π β§ π§ β π) β {π’ β πΆ β£ (π’ β© {π§}) β β
} β Fin) |
49 | | eleq2 2827 |
. . . . . 6
β’ (π¦ = {π§} β (π§ β π¦ β π§ β {π§})) |
50 | | ineq2 4171 |
. . . . . . . . 9
β’ (π¦ = {π§} β (π’ β© π¦) = (π’ β© {π§})) |
51 | 50 | neeq1d 3004 |
. . . . . . . 8
β’ (π¦ = {π§} β ((π’ β© π¦) β β
β (π’ β© {π§}) β β
)) |
52 | 51 | rabbidv 3418 |
. . . . . . 7
β’ (π¦ = {π§} β {π’ β πΆ β£ (π’ β© π¦) β β
} = {π’ β πΆ β£ (π’ β© {π§}) β β
}) |
53 | 52 | eleq1d 2823 |
. . . . . 6
β’ (π¦ = {π§} β ({π’ β πΆ β£ (π’ β© π¦) β β
} β Fin β {π’ β πΆ β£ (π’ β© {π§}) β β
} β
Fin)) |
54 | 49, 53 | anbi12d 632 |
. . . . 5
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 β
} β
Fin))) |
55 | 54 | rspcev 3584 |
. . . 4
β’ (({π§} β π« π β§ (π§ β {π§} β§ {π’ β πΆ β£ (π’ β© {π§}) β β
} β Fin)) β
βπ¦ β π«
π(π§ β π¦ β§ {π’ β πΆ β£ (π’ β© π¦) β β
} β Fin)) |
56 | 4, 6, 48, 55 | syl12anc 836 |
. . 3
β’ ((π β 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)) |
57 | 56 | ralrimiva 3144 |
. 2
β’ (π β π β βπ§ β π βπ¦ β π« π(π§ β π¦ β§ {π’ β πΆ β£ (π’ β© π¦) β β
} β Fin)) |
58 | | unipw 5412 |
. . . 4
β’ βͺ π« π = π |
59 | 58 | eqcomi 2746 |
. . 3
β’ π = βͺ
π« π |
60 | 10 | unisngl 22894 |
. . 3
β’ π = βͺ
πΆ |
61 | 59, 60 | islocfin 22884 |
. 2
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))) |
62 | 1, 2, 57, 61 | syl3anbrc 1344 |
1
β’ (π β π β πΆ β (LocFinβπ« π)) |