Metamath Proof Explorer


Theorem fsetdmprc0

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 } = (/) )

Proof

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 } = (/) )