Step |
Hyp |
Ref |
Expression |
1 |
|
clnbgr3stgrgrlic.n |
|- N e. NN0 |
2 |
|
clnbgr3stgrgrlic.v |
|- V = ( Vtx ` G ) |
3 |
|
clnbgr3stgrgrlic.w |
|- W = ( Vtx ` H ) |
4 |
2
|
fvexi |
|- V e. _V |
5 |
3
|
fvexi |
|- W e. _V |
6 |
4 5
|
pm3.2i |
|- ( V e. _V /\ W e. _V ) |
7 |
|
breng |
|- ( ( V e. _V /\ W e. _V ) -> ( V ~~ W <-> E. f f : V -1-1-onto-> W ) ) |
8 |
6 7
|
mp1i |
|- ( ( G e. USGraph /\ H e. USGraph ) -> ( V ~~ W <-> E. f f : V -1-1-onto-> W ) ) |
9 |
|
usgruhgr |
|- ( H e. USGraph -> H e. UHGraph ) |
10 |
9
|
adantl |
|- ( ( G e. USGraph /\ H e. USGraph ) -> H e. UHGraph ) |
11 |
10
|
3ad2ant1 |
|- ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) -> H e. UHGraph ) |
12 |
3
|
clnbgrssvtx |
|- ( H ClNeighbVtx ( f ` x ) ) C_ W |
13 |
12
|
a1i |
|- ( ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) /\ x e. V ) -> ( H ClNeighbVtx ( f ` x ) ) C_ W ) |
14 |
3
|
isubgruhgr |
|- ( ( H e. UHGraph /\ ( H ClNeighbVtx ( f ` x ) ) C_ W ) -> ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) e. UHGraph ) |
15 |
11 13 14
|
syl2an2r |
|- ( ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) /\ x e. V ) -> ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) e. UHGraph ) |
16 |
|
f1of |
|- ( f : V -1-1-onto-> W -> f : V --> W ) |
17 |
16
|
3ad2ant2 |
|- ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ x e. V ) -> f : V --> W ) |
18 |
|
simp3 |
|- ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ x e. V ) -> x e. V ) |
19 |
17 18
|
ffvelcdmd |
|- ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ x e. V ) -> ( f ` x ) e. W ) |
20 |
|
oveq2 |
|- ( y = ( f ` x ) -> ( H ClNeighbVtx y ) = ( H ClNeighbVtx ( f ` x ) ) ) |
21 |
20
|
oveq2d |
|- ( y = ( f ` x ) -> ( H ISubGr ( H ClNeighbVtx y ) ) = ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) |
22 |
21
|
breq1d |
|- ( y = ( f ` x ) -> ( ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) <-> ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ~=gr ( StarGr ` N ) ) ) |
23 |
22
|
rspcv |
|- ( ( f ` x ) e. W -> ( A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) -> ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ~=gr ( StarGr ` N ) ) ) |
24 |
19 23
|
syl |
|- ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ x e. V ) -> ( A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) -> ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ~=gr ( StarGr ` N ) ) ) |
25 |
24
|
3exp |
|- ( ( G e. USGraph /\ H e. USGraph ) -> ( f : V -1-1-onto-> W -> ( x e. V -> ( A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) -> ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ~=gr ( StarGr ` N ) ) ) ) ) |
26 |
25
|
com34 |
|- ( ( G e. USGraph /\ H e. USGraph ) -> ( f : V -1-1-onto-> W -> ( A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) -> ( x e. V -> ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ~=gr ( StarGr ` N ) ) ) ) ) |
27 |
26
|
3imp1 |
|- ( ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) /\ x e. V ) -> ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ~=gr ( StarGr ` N ) ) |
28 |
|
gricsym |
|- ( ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) e. UHGraph -> ( ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ~=gr ( StarGr ` N ) -> ( StarGr ` N ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) |
29 |
15 27 28
|
sylc |
|- ( ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) /\ x e. V ) -> ( StarGr ` N ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) |
30 |
29
|
anim1ci |
|- ( ( ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) /\ x e. V ) /\ ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) ) -> ( ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ ( StarGr ` N ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) |
31 |
|
grictr |
|- ( ( ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ ( StarGr ` N ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) -> ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) |
32 |
30 31
|
syl |
|- ( ( ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) /\ x e. V ) /\ ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) ) -> ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) |
33 |
32
|
ex |
|- ( ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) /\ x e. V ) -> ( ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) -> ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) |
34 |
33
|
ralimdva |
|- ( ( ( G e. USGraph /\ H e. USGraph ) /\ f : V -1-1-onto-> W /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) -> ( A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) -> A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) |
35 |
34
|
3exp |
|- ( ( G e. USGraph /\ H e. USGraph ) -> ( f : V -1-1-onto-> W -> ( A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) -> ( A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) -> A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) ) ) |
36 |
35
|
com24 |
|- ( ( G e. USGraph /\ H e. USGraph ) -> ( A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) -> ( A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) -> ( f : V -1-1-onto-> W -> A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) ) ) |
37 |
36
|
imp32 |
|- ( ( ( G e. USGraph /\ H e. USGraph ) /\ ( A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) ) -> ( f : V -1-1-onto-> W -> A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) |
38 |
37
|
ancld |
|- ( ( ( G e. USGraph /\ H e. USGraph ) /\ ( A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) ) -> ( f : V -1-1-onto-> W -> ( f : V -1-1-onto-> W /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) ) |
39 |
38
|
eximdv |
|- ( ( ( G e. USGraph /\ H e. USGraph ) /\ ( A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) ) -> ( E. f f : V -1-1-onto-> W -> E. f ( f : V -1-1-onto-> W /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) ) |
40 |
39
|
ex |
|- ( ( G e. USGraph /\ H e. USGraph ) -> ( ( A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) -> ( E. f f : V -1-1-onto-> W -> E. f ( f : V -1-1-onto-> W /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) ) ) |
41 |
40
|
com23 |
|- ( ( G e. USGraph /\ H e. USGraph ) -> ( E. f f : V -1-1-onto-> W -> ( ( A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) -> E. f ( f : V -1-1-onto-> W /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) ) ) |
42 |
8 41
|
sylbid |
|- ( ( G e. USGraph /\ H e. USGraph ) -> ( V ~~ W -> ( ( A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) -> E. f ( f : V -1-1-onto-> W /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) ) ) |
43 |
42
|
3impia |
|- ( ( G e. USGraph /\ H e. USGraph /\ V ~~ W ) -> ( ( A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) -> E. f ( f : V -1-1-onto-> W /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) ) |
44 |
43
|
3impib |
|- ( ( ( G e. USGraph /\ H e. USGraph /\ V ~~ W ) /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) -> E. f ( f : V -1-1-onto-> W /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) |
45 |
2 3
|
dfgrlic2 |
|- ( ( G e. USGraph /\ H e. USGraph ) -> ( G ~=lgr H <-> E. f ( f : V -1-1-onto-> W /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) ) |
46 |
45
|
3adant3 |
|- ( ( G e. USGraph /\ H e. USGraph /\ V ~~ W ) -> ( G ~=lgr H <-> E. f ( f : V -1-1-onto-> W /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) ) |
47 |
46
|
3ad2ant1 |
|- ( ( ( G e. USGraph /\ H e. USGraph /\ V ~~ W ) /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) -> ( G ~=lgr H <-> E. f ( f : V -1-1-onto-> W /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( H ISubGr ( H ClNeighbVtx ( f ` x ) ) ) ) ) ) |
48 |
44 47
|
mpbird |
|- ( ( ( G e. USGraph /\ H e. USGraph /\ V ~~ W ) /\ A. x e. V ( G ISubGr ( G ClNeighbVtx x ) ) ~=gr ( StarGr ` N ) /\ A. y e. W ( H ISubGr ( H ClNeighbVtx y ) ) ~=gr ( StarGr ` N ) ) -> G ~=lgr H ) |