Step |
Hyp |
Ref |
Expression |
1 |
|
clwwlk.v |
|- V = ( Vtx ` G ) |
2 |
|
clwwlk.e |
|- E = ( Edg ` G ) |
3 |
|
df-clwwlk |
|- ClWWalks = ( g e. _V |-> { w e. Word ( Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ( Edg ` g ) /\ { ( lastS ` w ) , ( w ` 0 ) } e. ( Edg ` g ) ) } ) |
4 |
|
fveq2 |
|- ( g = G -> ( Vtx ` g ) = ( Vtx ` G ) ) |
5 |
4 1
|
eqtr4di |
|- ( g = G -> ( Vtx ` g ) = V ) |
6 |
|
wrdeq |
|- ( ( Vtx ` g ) = V -> Word ( Vtx ` g ) = Word V ) |
7 |
5 6
|
syl |
|- ( g = G -> Word ( Vtx ` g ) = Word V ) |
8 |
|
fveq2 |
|- ( g = G -> ( Edg ` g ) = ( Edg ` G ) ) |
9 |
8 2
|
eqtr4di |
|- ( g = G -> ( Edg ` g ) = E ) |
10 |
9
|
eleq2d |
|- ( g = G -> ( { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ( Edg ` g ) <-> { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E ) ) |
11 |
10
|
ralbidv |
|- ( g = G -> ( A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ( Edg ` g ) <-> A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E ) ) |
12 |
9
|
eleq2d |
|- ( g = G -> ( { ( lastS ` w ) , ( w ` 0 ) } e. ( Edg ` g ) <-> { ( lastS ` w ) , ( w ` 0 ) } e. E ) ) |
13 |
11 12
|
3anbi23d |
|- ( g = G -> ( ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ( Edg ` g ) /\ { ( lastS ` w ) , ( w ` 0 ) } e. ( Edg ` g ) ) <-> ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) ) ) |
14 |
7 13
|
rabeqbidv |
|- ( g = G -> { w e. Word ( Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ( Edg ` g ) /\ { ( lastS ` w ) , ( w ` 0 ) } e. ( Edg ` g ) ) } = { w e. Word V | ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } ) |
15 |
|
id |
|- ( G e. _V -> G e. _V ) |
16 |
1
|
fvexi |
|- V e. _V |
17 |
16
|
a1i |
|- ( G e. _V -> V e. _V ) |
18 |
|
wrdexg |
|- ( V e. _V -> Word V e. _V ) |
19 |
|
rabexg |
|- ( Word V e. _V -> { w e. Word V | ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } e. _V ) |
20 |
17 18 19
|
3syl |
|- ( G e. _V -> { w e. Word V | ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } e. _V ) |
21 |
3 14 15 20
|
fvmptd3 |
|- ( G e. _V -> ( ClWWalks ` G ) = { w e. Word V | ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } ) |
22 |
|
fvprc |
|- ( -. G e. _V -> ( ClWWalks ` G ) = (/) ) |
23 |
|
noel |
|- -. { ( lastS ` w ) , ( w ` 0 ) } e. (/) |
24 |
|
fvprc |
|- ( -. G e. _V -> ( Edg ` G ) = (/) ) |
25 |
2 24
|
eqtrid |
|- ( -. G e. _V -> E = (/) ) |
26 |
25
|
eleq2d |
|- ( -. G e. _V -> ( { ( lastS ` w ) , ( w ` 0 ) } e. E <-> { ( lastS ` w ) , ( w ` 0 ) } e. (/) ) ) |
27 |
23 26
|
mtbiri |
|- ( -. G e. _V -> -. { ( lastS ` w ) , ( w ` 0 ) } e. E ) |
28 |
27
|
adantr |
|- ( ( -. G e. _V /\ w e. Word V ) -> -. { ( lastS ` w ) , ( w ` 0 ) } e. E ) |
29 |
28
|
intn3an3d |
|- ( ( -. G e. _V /\ w e. Word V ) -> -. ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) ) |
30 |
29
|
ralrimiva |
|- ( -. G e. _V -> A. w e. Word V -. ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) ) |
31 |
|
rabeq0 |
|- ( { w e. Word V | ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } = (/) <-> A. w e. Word V -. ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) ) |
32 |
30 31
|
sylibr |
|- ( -. G e. _V -> { w e. Word V | ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } = (/) ) |
33 |
22 32
|
eqtr4d |
|- ( -. G e. _V -> ( ClWWalks ` G ) = { w e. Word V | ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } ) |
34 |
21 33
|
pm2.61i |
|- ( ClWWalks ` G ) = { w e. Word V | ( w =/= (/) /\ A. i e. ( 0 ..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { ( lastS ` w ) , ( w ` 0 ) } e. E ) } |