Step | Hyp | Ref
| Expression |
1 | | frgrregorufr0.v |
. . 3
β’ π = (VtxβπΊ) |
2 | | frgrregorufr0.d |
. . 3
β’ π· = (VtxDegβπΊ) |
3 | | fveqeq2 6856 |
. . . 4
β’ (π₯ = π¦ β ((π·βπ₯) = πΎ β (π·βπ¦) = πΎ)) |
4 | 3 | cbvrabv 3420 |
. . 3
β’ {π₯ β π β£ (π·βπ₯) = πΎ} = {π¦ β π β£ (π·βπ¦) = πΎ} |
5 | | eqid 2737 |
. . 3
β’ (π β {π₯ β π β£ (π·βπ₯) = πΎ}) = (π β {π₯ β π β£ (π·βπ₯) = πΎ}) |
6 | 1, 2, 4, 5 | frgrwopreg 29309 |
. 2
β’ (πΊ β FriendGraph β
(((β―β{π₯ β
π β£ (π·βπ₯) = πΎ}) = 1 β¨ {π₯ β π β£ (π·βπ₯) = πΎ} = β
) β¨ ((β―β(π β {π₯ β π β£ (π·βπ₯) = πΎ})) = 1 β¨ (π β {π₯ β π β£ (π·βπ₯) = πΎ}) = β
))) |
7 | | frgrregorufr0.e |
. . . . . . 7
β’ πΈ = (EdgβπΊ) |
8 | 1, 2, 4, 5, 7 | frgrwopreg1 29304 |
. . . . . 6
β’ ((πΊ β FriendGraph β§
(β―β{π₯ β
π β£ (π·βπ₯) = πΎ}) = 1) β βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ) |
9 | 8 | 3mix3d 1339 |
. . . . 5
β’ ((πΊ β FriendGraph β§
(β―β{π₯ β
π β£ (π·βπ₯) = πΎ}) = 1) β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) |
10 | 9 | expcom 415 |
. . . 4
β’
((β―β{π₯
β π β£ (π·βπ₯) = πΎ}) = 1 β (πΊ β FriendGraph β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
11 | | fveqeq2 6856 |
. . . . . . . 8
β’ (π₯ = π£ β ((π·βπ₯) = πΎ β (π·βπ£) = πΎ)) |
12 | 11 | cbvrabv 3420 |
. . . . . . 7
β’ {π₯ β π β£ (π·βπ₯) = πΎ} = {π£ β π β£ (π·βπ£) = πΎ} |
13 | 12 | eqeq1i 2742 |
. . . . . 6
β’ ({π₯ β π β£ (π·βπ₯) = πΎ} = β
β {π£ β π β£ (π·βπ£) = πΎ} = β
) |
14 | | rabeq0 4349 |
. . . . . 6
β’ ({π£ β π β£ (π·βπ£) = πΎ} = β
β βπ£ β π Β¬ (π·βπ£) = πΎ) |
15 | 13, 14 | bitri 275 |
. . . . 5
β’ ({π₯ β π β£ (π·βπ₯) = πΎ} = β
β βπ£ β π Β¬ (π·βπ£) = πΎ) |
16 | | neqne 2952 |
. . . . . . . 8
β’ (Β¬
(π·βπ£) = πΎ β (π·βπ£) β πΎ) |
17 | 16 | ralimi 3087 |
. . . . . . 7
β’
(βπ£ β
π Β¬ (π·βπ£) = πΎ β βπ£ β π (π·βπ£) β πΎ) |
18 | 17 | 3mix2d 1338 |
. . . . . 6
β’
(βπ£ β
π Β¬ (π·βπ£) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) |
19 | 18 | a1d 25 |
. . . . 5
β’
(βπ£ β
π Β¬ (π·βπ£) = πΎ β (πΊ β FriendGraph β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
20 | 15, 19 | sylbi 216 |
. . . 4
β’ ({π₯ β π β£ (π·βπ₯) = πΎ} = β
β (πΊ β FriendGraph β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
21 | 10, 20 | jaoi 856 |
. . 3
β’
(((β―β{π₯
β π β£ (π·βπ₯) = πΎ}) = 1 β¨ {π₯ β π β£ (π·βπ₯) = πΎ} = β
) β (πΊ β FriendGraph β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
22 | 1, 2, 4, 5, 7 | frgrwopreg2 29305 |
. . . . . 6
β’ ((πΊ β FriendGraph β§
(β―β(π β
{π₯ β π β£ (π·βπ₯) = πΎ})) = 1) β βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ) |
23 | 22 | 3mix3d 1339 |
. . . . 5
β’ ((πΊ β FriendGraph β§
(β―β(π β
{π₯ β π β£ (π·βπ₯) = πΎ})) = 1) β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) |
24 | 23 | expcom 415 |
. . . 4
β’
((β―β(π
β {π₯ β π β£ (π·βπ₯) = πΎ})) = 1 β (πΊ β FriendGraph β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
25 | | difrab0eq 4434 |
. . . . 5
β’ ((π β {π₯ β π β£ (π·βπ₯) = πΎ}) = β
β π = {π₯ β π β£ (π·βπ₯) = πΎ}) |
26 | 12 | eqeq2i 2750 |
. . . . . . 7
β’ (π = {π₯ β π β£ (π·βπ₯) = πΎ} β π = {π£ β π β£ (π·βπ£) = πΎ}) |
27 | | rabid2 3439 |
. . . . . . 7
β’ (π = {π£ β π β£ (π·βπ£) = πΎ} β βπ£ β π (π·βπ£) = πΎ) |
28 | 26, 27 | bitri 275 |
. . . . . 6
β’ (π = {π₯ β π β£ (π·βπ₯) = πΎ} β βπ£ β π (π·βπ£) = πΎ) |
29 | | 3mix1 1331 |
. . . . . . 7
β’
(βπ£ β
π (π·βπ£) = πΎ β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) |
30 | 29 | a1d 25 |
. . . . . 6
β’
(βπ£ β
π (π·βπ£) = πΎ β (πΊ β FriendGraph β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
31 | 28, 30 | sylbi 216 |
. . . . 5
β’ (π = {π₯ β π β£ (π·βπ₯) = πΎ} β (πΊ β FriendGraph β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
32 | 25, 31 | sylbi 216 |
. . . 4
β’ ((π β {π₯ β π β£ (π·βπ₯) = πΎ}) = β
β (πΊ β FriendGraph β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
33 | 24, 32 | jaoi 856 |
. . 3
β’
(((β―β(π
β {π₯ β π β£ (π·βπ₯) = πΎ})) = 1 β¨ (π β {π₯ β π β£ (π·βπ₯) = πΎ}) = β
) β (πΊ β FriendGraph β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
34 | 21, 33 | jaoi 856 |
. 2
β’
((((β―β{π₯
β π β£ (π·βπ₯) = πΎ}) = 1 β¨ {π₯ β π β£ (π·βπ₯) = πΎ} = β
) β¨ ((β―β(π β {π₯ β π β£ (π·βπ₯) = πΎ})) = 1 β¨ (π β {π₯ β π β£ (π·βπ₯) = πΎ}) = β
)) β (πΊ β FriendGraph β (βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ))) |
35 | 6, 34 | mpcom 38 |
1
β’ (πΊ β FriendGraph β
(βπ£ β π (π·βπ£) = πΎ β¨ βπ£ β π (π·βπ£) β πΎ β¨ βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) |