| Step |
Hyp |
Ref |
Expression |
| 1 |
|
usgrexmpl.v |
|- V = ( 0 ... 4 ) |
| 2 |
|
usgrexmpl.e |
|- E = <" { 0 , 1 } { 1 , 2 } { 2 , 0 } { 0 , 3 } "> |
| 3 |
|
usgrexmpl.g |
|- G = <. V , E >. |
| 4 |
1
|
ovexi |
|- V e. _V |
| 5 |
|
s4cli |
|- <" { 0 , 1 } { 1 , 2 } { 2 , 0 } { 0 , 3 } "> e. Word _V |
| 6 |
5
|
elexi |
|- <" { 0 , 1 } { 1 , 2 } { 2 , 0 } { 0 , 3 } "> e. _V |
| 7 |
2 6
|
eqeltri |
|- E e. _V |
| 8 |
|
opvtxfv |
|- ( ( V e. _V /\ E e. _V ) -> ( Vtx ` <. V , E >. ) = V ) |
| 9 |
|
opiedgfv |
|- ( ( V e. _V /\ E e. _V ) -> ( iEdg ` <. V , E >. ) = E ) |
| 10 |
8 9
|
jca |
|- ( ( V e. _V /\ E e. _V ) -> ( ( Vtx ` <. V , E >. ) = V /\ ( iEdg ` <. V , E >. ) = E ) ) |
| 11 |
4 7 10
|
mp2an |
|- ( ( Vtx ` <. V , E >. ) = V /\ ( iEdg ` <. V , E >. ) = E ) |
| 12 |
3
|
fveq2i |
|- ( Vtx ` G ) = ( Vtx ` <. V , E >. ) |
| 13 |
12
|
eqeq1i |
|- ( ( Vtx ` G ) = V <-> ( Vtx ` <. V , E >. ) = V ) |
| 14 |
3
|
fveq2i |
|- ( iEdg ` G ) = ( iEdg ` <. V , E >. ) |
| 15 |
14
|
eqeq1i |
|- ( ( iEdg ` G ) = E <-> ( iEdg ` <. V , E >. ) = E ) |
| 16 |
13 15
|
anbi12i |
|- ( ( ( Vtx ` G ) = V /\ ( iEdg ` G ) = E ) <-> ( ( Vtx ` <. V , E >. ) = V /\ ( iEdg ` <. V , E >. ) = E ) ) |
| 17 |
11 16
|
mpbir |
|- ( ( Vtx ` G ) = V /\ ( iEdg ` G ) = E ) |