Step |
Hyp |
Ref |
Expression |
1 |
|
1egrvtxdg1.v |
|- ( ph -> ( Vtx ` G ) = V ) |
2 |
|
1egrvtxdg1.a |
|- ( ph -> A e. X ) |
3 |
|
1egrvtxdg1.b |
|- ( ph -> B e. V ) |
4 |
|
1egrvtxdg1.c |
|- ( ph -> C e. V ) |
5 |
|
1egrvtxdg1.n |
|- ( ph -> B =/= C ) |
6 |
|
1egrvtxdg1.i |
|- ( ph -> ( iEdg ` G ) = { <. A , { B , C } >. } ) |
7 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
8 |
3 1
|
eleqtrrd |
|- ( ph -> B e. ( Vtx ` G ) ) |
9 |
4 1
|
eleqtrrd |
|- ( ph -> C e. ( Vtx ` G ) ) |
10 |
7 2 8 9 6 5
|
usgr1e |
|- ( ph -> G e. USGraph ) |
11 |
|
eqid |
|- ( iEdg ` G ) = ( iEdg ` G ) |
12 |
|
eqid |
|- dom ( iEdg ` G ) = dom ( iEdg ` G ) |
13 |
|
eqid |
|- ( VtxDeg ` G ) = ( VtxDeg ` G ) |
14 |
7 11 12 13
|
vtxdusgrval |
|- ( ( G e. USGraph /\ B e. ( Vtx ` G ) ) -> ( ( VtxDeg ` G ) ` B ) = ( # ` { x e. dom ( iEdg ` G ) | B e. ( ( iEdg ` G ) ` x ) } ) ) |
15 |
10 8 14
|
syl2anc |
|- ( ph -> ( ( VtxDeg ` G ) ` B ) = ( # ` { x e. dom ( iEdg ` G ) | B e. ( ( iEdg ` G ) ` x ) } ) ) |
16 |
|
dmeq |
|- ( ( iEdg ` G ) = { <. A , { B , C } >. } -> dom ( iEdg ` G ) = dom { <. A , { B , C } >. } ) |
17 |
16
|
adantl |
|- ( ( ph /\ ( iEdg ` G ) = { <. A , { B , C } >. } ) -> dom ( iEdg ` G ) = dom { <. A , { B , C } >. } ) |
18 |
|
prex |
|- { B , C } e. _V |
19 |
|
dmsnopg |
|- ( { B , C } e. _V -> dom { <. A , { B , C } >. } = { A } ) |
20 |
18 19
|
mp1i |
|- ( ( ph /\ ( iEdg ` G ) = { <. A , { B , C } >. } ) -> dom { <. A , { B , C } >. } = { A } ) |
21 |
17 20
|
eqtrd |
|- ( ( ph /\ ( iEdg ` G ) = { <. A , { B , C } >. } ) -> dom ( iEdg ` G ) = { A } ) |
22 |
|
fveq1 |
|- ( ( iEdg ` G ) = { <. A , { B , C } >. } -> ( ( iEdg ` G ) ` x ) = ( { <. A , { B , C } >. } ` x ) ) |
23 |
22
|
eleq2d |
|- ( ( iEdg ` G ) = { <. A , { B , C } >. } -> ( B e. ( ( iEdg ` G ) ` x ) <-> B e. ( { <. A , { B , C } >. } ` x ) ) ) |
24 |
23
|
adantl |
|- ( ( ph /\ ( iEdg ` G ) = { <. A , { B , C } >. } ) -> ( B e. ( ( iEdg ` G ) ` x ) <-> B e. ( { <. A , { B , C } >. } ` x ) ) ) |
25 |
21 24
|
rabeqbidv |
|- ( ( ph /\ ( iEdg ` G ) = { <. A , { B , C } >. } ) -> { x e. dom ( iEdg ` G ) | B e. ( ( iEdg ` G ) ` x ) } = { x e. { A } | B e. ( { <. A , { B , C } >. } ` x ) } ) |
26 |
25
|
fveq2d |
|- ( ( ph /\ ( iEdg ` G ) = { <. A , { B , C } >. } ) -> ( # ` { x e. dom ( iEdg ` G ) | B e. ( ( iEdg ` G ) ` x ) } ) = ( # ` { x e. { A } | B e. ( { <. A , { B , C } >. } ` x ) } ) ) |
27 |
|
fveq2 |
|- ( x = A -> ( { <. A , { B , C } >. } ` x ) = ( { <. A , { B , C } >. } ` A ) ) |
28 |
27
|
eleq2d |
|- ( x = A -> ( B e. ( { <. A , { B , C } >. } ` x ) <-> B e. ( { <. A , { B , C } >. } ` A ) ) ) |
29 |
28
|
rabsnif |
|- { x e. { A } | B e. ( { <. A , { B , C } >. } ` x ) } = if ( B e. ( { <. A , { B , C } >. } ` A ) , { A } , (/) ) |
30 |
|
prid1g |
|- ( B e. V -> B e. { B , C } ) |
31 |
3 30
|
syl |
|- ( ph -> B e. { B , C } ) |
32 |
|
fvsng |
|- ( ( A e. X /\ { B , C } e. _V ) -> ( { <. A , { B , C } >. } ` A ) = { B , C } ) |
33 |
2 18 32
|
sylancl |
|- ( ph -> ( { <. A , { B , C } >. } ` A ) = { B , C } ) |
34 |
31 33
|
eleqtrrd |
|- ( ph -> B e. ( { <. A , { B , C } >. } ` A ) ) |
35 |
34
|
iftrued |
|- ( ph -> if ( B e. ( { <. A , { B , C } >. } ` A ) , { A } , (/) ) = { A } ) |
36 |
29 35
|
syl5eq |
|- ( ph -> { x e. { A } | B e. ( { <. A , { B , C } >. } ` x ) } = { A } ) |
37 |
36
|
fveq2d |
|- ( ph -> ( # ` { x e. { A } | B e. ( { <. A , { B , C } >. } ` x ) } ) = ( # ` { A } ) ) |
38 |
|
hashsng |
|- ( A e. X -> ( # ` { A } ) = 1 ) |
39 |
2 38
|
syl |
|- ( ph -> ( # ` { A } ) = 1 ) |
40 |
37 39
|
eqtrd |
|- ( ph -> ( # ` { x e. { A } | B e. ( { <. A , { B , C } >. } ` x ) } ) = 1 ) |
41 |
40
|
adantr |
|- ( ( ph /\ ( iEdg ` G ) = { <. A , { B , C } >. } ) -> ( # ` { x e. { A } | B e. ( { <. A , { B , C } >. } ` x ) } ) = 1 ) |
42 |
26 41
|
eqtrd |
|- ( ( ph /\ ( iEdg ` G ) = { <. A , { B , C } >. } ) -> ( # ` { x e. dom ( iEdg ` G ) | B e. ( ( iEdg ` G ) ` x ) } ) = 1 ) |
43 |
6 42
|
mpdan |
|- ( ph -> ( # ` { x e. dom ( iEdg ` G ) | B e. ( ( iEdg ` G ) ` x ) } ) = 1 ) |
44 |
15 43
|
eqtrd |
|- ( ph -> ( ( VtxDeg ` G ) ` B ) = 1 ) |