Description: The set of functions with a proper class as domain is empty. (Contributed by AV, 22-Aug-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | fsetdmprc0 | |- ( A e/ _V -> { f | f Fn A } = (/) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel | |- ( A e/ _V <-> -. A e. _V ) |
|
2 | vex | |- f e. _V |
|
3 | 2 | a1i | |- ( f Fn A -> f e. _V ) |
4 | id | |- ( f Fn A -> f Fn A ) |
|
5 | 3 4 | fndmexd | |- ( f Fn A -> A e. _V ) |
6 | 5 | con3i | |- ( -. A e. _V -> -. f Fn A ) |
7 | 1 6 | sylbi | |- ( A e/ _V -> -. f Fn A ) |
8 | 7 | alrimiv | |- ( A e/ _V -> A. f -. f Fn A ) |
9 | ab0 | |- ( { f | f Fn A } = (/) <-> A. f -. f Fn A ) |
|
10 | 8 9 | sylibr | |- ( A e/ _V -> { f | f Fn A } = (/) ) |