Metamath Proof Explorer

Theorem ssdmres

Description: A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997)

Ref Expression
Assertion ssdmres 
|- ( A C_ dom B <-> dom ( B |` A ) = A )

Proof

Step Hyp Ref Expression
1 df-ss 
 |-  ( A C_ dom B <-> ( A i^i dom B ) = A )
2 dmres 
 |-  dom ( B |` A ) = ( A i^i dom B )
3 2 eqeq1i 
 |-  ( dom ( B |` A ) = A <-> ( A i^i dom B ) = A )
4 1 3 bitr4i 
 |-  ( A C_ dom B <-> dom ( B |` A ) = A )