Step |
Hyp |
Ref |
Expression |
1 |
|
isumgr.v |
|- V = ( Vtx ` G ) |
2 |
|
isumgr.e |
|- E = ( iEdg ` G ) |
3 |
1 2
|
isumgrs |
|- ( G e. U -> ( G e. UMGraph <-> E : dom E --> { x e. ~P V | ( # ` x ) = 2 } ) ) |
4 |
3
|
adantr |
|- ( ( G e. U /\ E e. Word X ) -> ( G e. UMGraph <-> E : dom E --> { x e. ~P V | ( # ` x ) = 2 } ) ) |
5 |
|
wrdf |
|- ( E e. Word X -> E : ( 0 ..^ ( # ` E ) ) --> X ) |
6 |
5
|
adantl |
|- ( ( G e. U /\ E e. Word X ) -> E : ( 0 ..^ ( # ` E ) ) --> X ) |
7 |
6
|
fdmd |
|- ( ( G e. U /\ E e. Word X ) -> dom E = ( 0 ..^ ( # ` E ) ) ) |
8 |
7
|
feq2d |
|- ( ( G e. U /\ E e. Word X ) -> ( E : dom E --> { x e. ~P V | ( # ` x ) = 2 } <-> E : ( 0 ..^ ( # ` E ) ) --> { x e. ~P V | ( # ` x ) = 2 } ) ) |
9 |
|
iswrdi |
|- ( E : ( 0 ..^ ( # ` E ) ) --> { x e. ~P V | ( # ` x ) = 2 } -> E e. Word { x e. ~P V | ( # ` x ) = 2 } ) |
10 |
|
wrdf |
|- ( E e. Word { x e. ~P V | ( # ` x ) = 2 } -> E : ( 0 ..^ ( # ` E ) ) --> { x e. ~P V | ( # ` x ) = 2 } ) |
11 |
9 10
|
impbii |
|- ( E : ( 0 ..^ ( # ` E ) ) --> { x e. ~P V | ( # ` x ) = 2 } <-> E e. Word { x e. ~P V | ( # ` x ) = 2 } ) |
12 |
8 11
|
bitrdi |
|- ( ( G e. U /\ E e. Word X ) -> ( E : dom E --> { x e. ~P V | ( # ` x ) = 2 } <-> E e. Word { x e. ~P V | ( # ` x ) = 2 } ) ) |
13 |
4 12
|
bitrd |
|- ( ( G e. U /\ E e. Word X ) -> ( G e. UMGraph <-> E e. Word { x e. ~P V | ( # ` x ) = 2 } ) ) |