Metamath Proof Explorer

Theorem fvpr0o

Description: The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023)

Ref Expression
Assertion fvpr0o 
|- ( A e. V -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )

Proof

Step Hyp Ref Expression
1 peano1 
 |-  (/) e. _om
2 1n0 
 |-  1o =/= (/)
3 2 necomi 
 |-  (/) =/= 1o
4 fvpr1g 
 |-  ( ( (/) e. _om /\ A e. V /\ (/) =/= 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
5 1 3 4 mp3an13 
 |-  ( A e. V -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )