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 | |- g e. _V |
|
3 | 2 | a1i | |- ( g Fn A -> g e. _V ) |
4 | id | |- ( g Fn A -> g Fn A ) |
|
5 | 3 4 | fndmexd | |- ( g Fn A -> A e. _V ) |
6 | 5 | con3i | |- ( -. A e. _V -> -. g Fn A ) |
7 | 1 6 | sylbi | |- ( A e/ _V -> -. g Fn A ) |
8 | 7 | alrimiv | |- ( A e/ _V -> A. g -. g Fn A ) |
9 | fneq1 | |- ( f = g -> ( f Fn A <-> g Fn A ) ) |
|
10 | 9 | ab0w | |- ( { f | f Fn A } = (/) <-> A. g -. g Fn A ) |
11 | 8 10 | sylibr | |- ( A e/ _V -> { f | f Fn A } = (/) ) |