Step |
Hyp |
Ref |
Expression |
1 |
|
f1dm |
|- ( F : A -1-1-> B -> dom F = A ) |
2 |
|
dmexg |
|- ( F e. V -> dom F e. _V ) |
3 |
|
eleq1 |
|- ( A = dom F -> ( A e. _V <-> dom F e. _V ) ) |
4 |
3
|
eqcoms |
|- ( dom F = A -> ( A e. _V <-> dom F e. _V ) ) |
5 |
2 4
|
syl5ibr |
|- ( dom F = A -> ( F e. V -> A e. _V ) ) |
6 |
1 5
|
syl |
|- ( F : A -1-1-> B -> ( F e. V -> A e. _V ) ) |
7 |
6
|
impcom |
|- ( ( F e. V /\ F : A -1-1-> B ) -> A e. _V ) |
8 |
|
f1dmvrnfibi |
|- ( ( A e. _V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) ) |
9 |
7 8
|
sylancom |
|- ( ( F e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) ) |