Step |
Hyp |
Ref |
Expression |
1 |
|
wwlks2onv.v |
|- V = ( Vtx ` G ) |
2 |
|
id |
|- ( W e. ( A ( 2 WWalksNOn G ) C ) -> W e. ( A ( 2 WWalksNOn G ) C ) ) |
3 |
1
|
elwwlks2ons3im |
|- ( W e. ( A ( 2 WWalksNOn G ) C ) -> ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) ) |
4 |
|
anass |
|- ( ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) <-> ( W e. ( A ( 2 WWalksNOn G ) C ) /\ ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) ) ) |
5 |
2 3 4
|
sylanbrc |
|- ( W e. ( A ( 2 WWalksNOn G ) C ) -> ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) ) |
6 |
|
simpr |
|- ( ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) -> ( W ` 1 ) e. V ) |
7 |
|
s3eq2 |
|- ( b = ( W ` 1 ) -> <" A b C "> = <" A ( W ` 1 ) C "> ) |
8 |
|
eqeq2 |
|- ( <" A b C "> = <" A ( W ` 1 ) C "> -> ( W = <" A b C "> <-> W = <" A ( W ` 1 ) C "> ) ) |
9 |
|
eleq1 |
|- ( <" A b C "> = <" A ( W ` 1 ) C "> -> ( <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) <-> <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) |
10 |
8 9
|
anbi12d |
|- ( <" A b C "> = <" A ( W ` 1 ) C "> -> ( ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) <-> ( W = <" A ( W ` 1 ) C "> /\ <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) ) |
11 |
7 10
|
syl |
|- ( b = ( W ` 1 ) -> ( ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) <-> ( W = <" A ( W ` 1 ) C "> /\ <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) ) |
12 |
11
|
adantl |
|- ( ( ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) /\ b = ( W ` 1 ) ) -> ( ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) <-> ( W = <" A ( W ` 1 ) C "> /\ <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) ) |
13 |
|
simpr |
|- ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) -> W = <" A ( W ` 1 ) C "> ) |
14 |
|
eleq1 |
|- ( W = <" A ( W ` 1 ) C "> -> ( W e. ( A ( 2 WWalksNOn G ) C ) <-> <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) |
15 |
14
|
biimpac |
|- ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) -> <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) |
16 |
13 15
|
jca |
|- ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) -> ( W = <" A ( W ` 1 ) C "> /\ <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) |
17 |
16
|
adantr |
|- ( ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) -> ( W = <" A ( W ` 1 ) C "> /\ <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) |
18 |
6 12 17
|
rspcedvd |
|- ( ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) -> E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) |
19 |
5 18
|
syl |
|- ( W e. ( A ( 2 WWalksNOn G ) C ) -> E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) |
20 |
|
eleq1 |
|- ( <" A b C "> = W -> ( <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) <-> W e. ( A ( 2 WWalksNOn G ) C ) ) ) |
21 |
20
|
eqcoms |
|- ( W = <" A b C "> -> ( <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) <-> W e. ( A ( 2 WWalksNOn G ) C ) ) ) |
22 |
21
|
biimpa |
|- ( ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) -> W e. ( A ( 2 WWalksNOn G ) C ) ) |
23 |
22
|
rexlimivw |
|- ( E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) -> W e. ( A ( 2 WWalksNOn G ) C ) ) |
24 |
19 23
|
impbii |
|- ( W e. ( A ( 2 WWalksNOn G ) C ) <-> E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) |