Step |
Hyp |
Ref |
Expression |
1 |
|
uzf |
|- ZZ>= : ZZ --> ~P ZZ |
2 |
|
ffn |
|- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ ) |
3 |
|
fvelrnb |
|- ( ZZ>= Fn ZZ -> ( M e. ran ZZ>= <-> E. k e. ZZ ( ZZ>= ` k ) = M ) ) |
4 |
1 2 3
|
mp2b |
|- ( M e. ran ZZ>= <-> E. k e. ZZ ( ZZ>= ` k ) = M ) |
5 |
|
uzid |
|- ( k e. ZZ -> k e. ( ZZ>= ` k ) ) |
6 |
5
|
ne0d |
|- ( k e. ZZ -> ( ZZ>= ` k ) =/= (/) ) |
7 |
|
neeq1 |
|- ( ( ZZ>= ` k ) = M -> ( ( ZZ>= ` k ) =/= (/) <-> M =/= (/) ) ) |
8 |
6 7
|
syl5ibcom |
|- ( k e. ZZ -> ( ( ZZ>= ` k ) = M -> M =/= (/) ) ) |
9 |
8
|
rexlimiv |
|- ( E. k e. ZZ ( ZZ>= ` k ) = M -> M =/= (/) ) |
10 |
4 9
|
sylbi |
|- ( M e. ran ZZ>= -> M =/= (/) ) |