Metamath Proof Explorer

Theorem nelrnres

Description: If A is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Assertion nelrnres 
|- ( -. A e. ran B -> -. A e. ran ( B |` C ) )

Proof

Step Hyp Ref Expression
1 rnresss 
 |-  ran ( B |` C ) C_ ran B
2 ssnel 
 |-  ( ( ran ( B |` C ) C_ ran B /\ -. A e. ran B ) -> -. A e. ran ( B |` C ) )
3 1 2 mpan 
 |-  ( -. A e. ran B -> -. A e. ran ( B |` C ) )