Step |
Hyp |
Ref |
Expression |
1 |
|
vtxdgfusgrf.v |
|- V = ( Vtx ` G ) |
2 |
|
fusgrfis |
|- ( G e. FinUSGraph -> ( Edg ` G ) e. Fin ) |
3 |
|
fusgrusgr |
|- ( G e. FinUSGraph -> G e. USGraph ) |
4 |
|
eqid |
|- ( iEdg ` G ) = ( iEdg ` G ) |
5 |
|
eqid |
|- ( Edg ` G ) = ( Edg ` G ) |
6 |
4 5
|
usgredgffibi |
|- ( G e. USGraph -> ( ( Edg ` G ) e. Fin <-> ( iEdg ` G ) e. Fin ) ) |
7 |
3 6
|
syl |
|- ( G e. FinUSGraph -> ( ( Edg ` G ) e. Fin <-> ( iEdg ` G ) e. Fin ) ) |
8 |
|
usgrfun |
|- ( G e. USGraph -> Fun ( iEdg ` G ) ) |
9 |
|
fundmfibi |
|- ( Fun ( iEdg ` G ) -> ( ( iEdg ` G ) e. Fin <-> dom ( iEdg ` G ) e. Fin ) ) |
10 |
3 8 9
|
3syl |
|- ( G e. FinUSGraph -> ( ( iEdg ` G ) e. Fin <-> dom ( iEdg ` G ) e. Fin ) ) |
11 |
7 10
|
bitrd |
|- ( G e. FinUSGraph -> ( ( Edg ` G ) e. Fin <-> dom ( iEdg ` G ) e. Fin ) ) |
12 |
2 11
|
mpbid |
|- ( G e. FinUSGraph -> dom ( iEdg ` G ) e. Fin ) |
13 |
|
eqid |
|- dom ( iEdg ` G ) = dom ( iEdg ` G ) |
14 |
1 4 13
|
vtxdgfisf |
|- ( ( G e. FinUSGraph /\ dom ( iEdg ` G ) e. Fin ) -> ( VtxDeg ` G ) : V --> NN0 ) |
15 |
12 14
|
mpdan |
|- ( G e. FinUSGraph -> ( VtxDeg ` G ) : V --> NN0 ) |