Step |
Hyp |
Ref |
Expression |
1 |
|
vtxdgf.v |
|- V = ( Vtx ` G ) |
2 |
|
vtxdg0e.i |
|- I = ( iEdg ` G ) |
3 |
2
|
eqeq1i |
|- ( I = (/) <-> ( iEdg ` G ) = (/) ) |
4 |
|
dmeq |
|- ( ( iEdg ` G ) = (/) -> dom ( iEdg ` G ) = dom (/) ) |
5 |
|
dm0 |
|- dom (/) = (/) |
6 |
4 5
|
eqtrdi |
|- ( ( iEdg ` G ) = (/) -> dom ( iEdg ` G ) = (/) ) |
7 |
|
0fin |
|- (/) e. Fin |
8 |
6 7
|
eqeltrdi |
|- ( ( iEdg ` G ) = (/) -> dom ( iEdg ` G ) e. Fin ) |
9 |
3 8
|
sylbi |
|- ( I = (/) -> dom ( iEdg ` G ) e. Fin ) |
10 |
|
simpl |
|- ( ( U e. V /\ I = (/) ) -> U e. V ) |
11 |
|
eqid |
|- ( iEdg ` G ) = ( iEdg ` G ) |
12 |
|
eqid |
|- dom ( iEdg ` G ) = dom ( iEdg ` G ) |
13 |
1 11 12
|
vtxdgfival |
|- ( ( dom ( iEdg ` G ) e. Fin /\ U e. V ) -> ( ( VtxDeg ` G ) ` U ) = ( ( # ` { x e. dom ( iEdg ` G ) | U e. ( ( iEdg ` G ) ` x ) } ) + ( # ` { x e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` x ) = { U } } ) ) ) |
14 |
9 10 13
|
syl2an2 |
|- ( ( U e. V /\ I = (/) ) -> ( ( VtxDeg ` G ) ` U ) = ( ( # ` { x e. dom ( iEdg ` G ) | U e. ( ( iEdg ` G ) ` x ) } ) + ( # ` { x e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` x ) = { U } } ) ) ) |
15 |
3 6
|
sylbi |
|- ( I = (/) -> dom ( iEdg ` G ) = (/) ) |
16 |
15
|
adantl |
|- ( ( U e. V /\ I = (/) ) -> dom ( iEdg ` G ) = (/) ) |
17 |
|
rabeq |
|- ( dom ( iEdg ` G ) = (/) -> { x e. dom ( iEdg ` G ) | U e. ( ( iEdg ` G ) ` x ) } = { x e. (/) | U e. ( ( iEdg ` G ) ` x ) } ) |
18 |
|
rab0 |
|- { x e. (/) | U e. ( ( iEdg ` G ) ` x ) } = (/) |
19 |
17 18
|
eqtrdi |
|- ( dom ( iEdg ` G ) = (/) -> { x e. dom ( iEdg ` G ) | U e. ( ( iEdg ` G ) ` x ) } = (/) ) |
20 |
19
|
fveq2d |
|- ( dom ( iEdg ` G ) = (/) -> ( # ` { x e. dom ( iEdg ` G ) | U e. ( ( iEdg ` G ) ` x ) } ) = ( # ` (/) ) ) |
21 |
|
hash0 |
|- ( # ` (/) ) = 0 |
22 |
20 21
|
eqtrdi |
|- ( dom ( iEdg ` G ) = (/) -> ( # ` { x e. dom ( iEdg ` G ) | U e. ( ( iEdg ` G ) ` x ) } ) = 0 ) |
23 |
|
rabeq |
|- ( dom ( iEdg ` G ) = (/) -> { x e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` x ) = { U } } = { x e. (/) | ( ( iEdg ` G ) ` x ) = { U } } ) |
24 |
23
|
fveq2d |
|- ( dom ( iEdg ` G ) = (/) -> ( # ` { x e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` x ) = { U } } ) = ( # ` { x e. (/) | ( ( iEdg ` G ) ` x ) = { U } } ) ) |
25 |
|
rab0 |
|- { x e. (/) | ( ( iEdg ` G ) ` x ) = { U } } = (/) |
26 |
25
|
fveq2i |
|- ( # ` { x e. (/) | ( ( iEdg ` G ) ` x ) = { U } } ) = ( # ` (/) ) |
27 |
26 21
|
eqtri |
|- ( # ` { x e. (/) | ( ( iEdg ` G ) ` x ) = { U } } ) = 0 |
28 |
24 27
|
eqtrdi |
|- ( dom ( iEdg ` G ) = (/) -> ( # ` { x e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` x ) = { U } } ) = 0 ) |
29 |
22 28
|
oveq12d |
|- ( dom ( iEdg ` G ) = (/) -> ( ( # ` { x e. dom ( iEdg ` G ) | U e. ( ( iEdg ` G ) ` x ) } ) + ( # ` { x e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` x ) = { U } } ) ) = ( 0 + 0 ) ) |
30 |
16 29
|
syl |
|- ( ( U e. V /\ I = (/) ) -> ( ( # ` { x e. dom ( iEdg ` G ) | U e. ( ( iEdg ` G ) ` x ) } ) + ( # ` { x e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` x ) = { U } } ) ) = ( 0 + 0 ) ) |
31 |
|
00id |
|- ( 0 + 0 ) = 0 |
32 |
30 31
|
eqtrdi |
|- ( ( U e. V /\ I = (/) ) -> ( ( # ` { x e. dom ( iEdg ` G ) | U e. ( ( iEdg ` G ) ` x ) } ) + ( # ` { x e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` x ) = { U } } ) ) = 0 ) |
33 |
14 32
|
eqtrd |
|- ( ( U e. V /\ I = (/) ) -> ( ( VtxDeg ` G ) ` U ) = 0 ) |