Step |
Hyp |
Ref |
Expression |
1 |
|
umgr2v2evtx.g |
|- G = <. V , { <. 0 , { A , B } >. , <. 1 , { A , B } >. } >. |
2 |
1
|
umgr2v2e |
|- ( ( ( V e. W /\ A e. V /\ B e. V ) /\ A =/= B ) -> G e. UMGraph ) |
3 |
1
|
umgr2v2evtxel |
|- ( ( V e. W /\ A e. V ) -> A e. ( Vtx ` G ) ) |
4 |
3
|
3adant3 |
|- ( ( V e. W /\ A e. V /\ B e. V ) -> A e. ( Vtx ` G ) ) |
5 |
4
|
adantr |
|- ( ( ( V e. W /\ A e. V /\ B e. V ) /\ A =/= B ) -> A e. ( Vtx ` G ) ) |
6 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
7 |
|
eqid |
|- ( iEdg ` G ) = ( iEdg ` G ) |
8 |
|
eqid |
|- dom ( iEdg ` G ) = dom ( iEdg ` G ) |
9 |
|
eqid |
|- ( VtxDeg ` G ) = ( VtxDeg ` G ) |
10 |
6 7 8 9
|
vtxdumgrval |
|- ( ( G e. UMGraph /\ A e. ( Vtx ` G ) ) -> ( ( VtxDeg ` G ) ` A ) = ( # ` { x e. dom ( iEdg ` G ) | A e. ( ( iEdg ` G ) ` x ) } ) ) |
11 |
2 5 10
|
syl2anc |
|- ( ( ( V e. W /\ A e. V /\ B e. V ) /\ A =/= B ) -> ( ( VtxDeg ` G ) ` A ) = ( # ` { x e. dom ( iEdg ` G ) | A e. ( ( iEdg ` G ) ` x ) } ) ) |
12 |
1
|
umgr2v2eiedg |
|- ( ( V e. W /\ A e. V /\ B e. V ) -> ( iEdg ` G ) = { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ) |
13 |
12
|
dmeqd |
|- ( ( V e. W /\ A e. V /\ B e. V ) -> dom ( iEdg ` G ) = dom { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ) |
14 |
|
prex |
|- { A , B } e. _V |
15 |
14 14
|
dmprop |
|- dom { <. 0 , { A , B } >. , <. 1 , { A , B } >. } = { 0 , 1 } |
16 |
13 15
|
eqtrdi |
|- ( ( V e. W /\ A e. V /\ B e. V ) -> dom ( iEdg ` G ) = { 0 , 1 } ) |
17 |
12
|
fveq1d |
|- ( ( V e. W /\ A e. V /\ B e. V ) -> ( ( iEdg ` G ) ` x ) = ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) ) |
18 |
17
|
eleq2d |
|- ( ( V e. W /\ A e. V /\ B e. V ) -> ( A e. ( ( iEdg ` G ) ` x ) <-> A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) ) ) |
19 |
16 18
|
rabeqbidv |
|- ( ( V e. W /\ A e. V /\ B e. V ) -> { x e. dom ( iEdg ` G ) | A e. ( ( iEdg ` G ) ` x ) } = { x e. { 0 , 1 } | A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) } ) |
20 |
19
|
fveq2d |
|- ( ( V e. W /\ A e. V /\ B e. V ) -> ( # ` { x e. dom ( iEdg ` G ) | A e. ( ( iEdg ` G ) ` x ) } ) = ( # ` { x e. { 0 , 1 } | A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) } ) ) |
21 |
|
prid1g |
|- ( A e. V -> A e. { A , B } ) |
22 |
|
0ne1 |
|- 0 =/= 1 |
23 |
|
c0ex |
|- 0 e. _V |
24 |
23 14
|
fvpr1 |
|- ( 0 =/= 1 -> ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 0 ) = { A , B } ) |
25 |
22 24
|
ax-mp |
|- ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 0 ) = { A , B } |
26 |
21 25
|
eleqtrrdi |
|- ( A e. V -> A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 0 ) ) |
27 |
|
1ex |
|- 1 e. _V |
28 |
27 14
|
fvpr2 |
|- ( 0 =/= 1 -> ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 1 ) = { A , B } ) |
29 |
22 28
|
ax-mp |
|- ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 1 ) = { A , B } |
30 |
21 29
|
eleqtrrdi |
|- ( A e. V -> A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 1 ) ) |
31 |
|
fveq2 |
|- ( x = 0 -> ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) = ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 0 ) ) |
32 |
31
|
eleq2d |
|- ( x = 0 -> ( A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) <-> A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 0 ) ) ) |
33 |
|
fveq2 |
|- ( x = 1 -> ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) = ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 1 ) ) |
34 |
33
|
eleq2d |
|- ( x = 1 -> ( A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) <-> A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 1 ) ) ) |
35 |
23 27 32 34
|
ralpr |
|- ( A. x e. { 0 , 1 } A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) <-> ( A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 0 ) /\ A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` 1 ) ) ) |
36 |
26 30 35
|
sylanbrc |
|- ( A e. V -> A. x e. { 0 , 1 } A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) ) |
37 |
|
rabid2 |
|- ( { 0 , 1 } = { x e. { 0 , 1 } | A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) } <-> A. x e. { 0 , 1 } A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) ) |
38 |
36 37
|
sylibr |
|- ( A e. V -> { 0 , 1 } = { x e. { 0 , 1 } | A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) } ) |
39 |
38
|
eqcomd |
|- ( A e. V -> { x e. { 0 , 1 } | A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) } = { 0 , 1 } ) |
40 |
39
|
fveq2d |
|- ( A e. V -> ( # ` { x e. { 0 , 1 } | A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) } ) = ( # ` { 0 , 1 } ) ) |
41 |
|
prhash2ex |
|- ( # ` { 0 , 1 } ) = 2 |
42 |
40 41
|
eqtrdi |
|- ( A e. V -> ( # ` { x e. { 0 , 1 } | A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) } ) = 2 ) |
43 |
42
|
3ad2ant2 |
|- ( ( V e. W /\ A e. V /\ B e. V ) -> ( # ` { x e. { 0 , 1 } | A e. ( { <. 0 , { A , B } >. , <. 1 , { A , B } >. } ` x ) } ) = 2 ) |
44 |
20 43
|
eqtrd |
|- ( ( V e. W /\ A e. V /\ B e. V ) -> ( # ` { x e. dom ( iEdg ` G ) | A e. ( ( iEdg ` G ) ` x ) } ) = 2 ) |
45 |
44
|
adantr |
|- ( ( ( V e. W /\ A e. V /\ B e. V ) /\ A =/= B ) -> ( # ` { x e. dom ( iEdg ` G ) | A e. ( ( iEdg ` G ) ` x ) } ) = 2 ) |
46 |
11 45
|
eqtrd |
|- ( ( ( V e. W /\ A e. V /\ B e. V ) /\ A =/= B ) -> ( ( VtxDeg ` G ) ` A ) = 2 ) |