Metamath Proof Explorer

Theorem relssres

Description: Simplification law for restriction. (Contributed by NM, 16-Aug-1994)

Ref Expression
Assertion relssres 
|- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( Rel A /\ dom A C_ B ) -> Rel A )
2 vex 
 |-  x e. _V
3 vex 
 |-  y e. _V
4 2 3 opeldm 
 |-  ( <. x , y >. e. A -> x e. dom A )
5 ssel 
 |-  ( dom A C_ B -> ( x e. dom A -> x e. B ) )
6 4 5 syl5 
 |-  ( dom A C_ B -> ( <. x , y >. e. A -> x e. B ) )
7 6 ancrd 
 |-  ( dom A C_ B -> ( <. x , y >. e. A -> ( x e. B /\ <. x , y >. e. A ) ) )
8 3 opelresi 
 |-  ( <. x , y >. e. ( A |` B ) <-> ( x e. B /\ <. x , y >. e. A ) )
9 7 8 syl6ibr 
 |-  ( dom A C_ B -> ( <. x , y >. e. A -> <. x , y >. e. ( A |` B ) ) )
10 9 adantl 
 |-  ( ( Rel A /\ dom A C_ B ) -> ( <. x , y >. e. A -> <. x , y >. e. ( A |` B ) ) )
11 1 10 relssdv 
 |-  ( ( Rel A /\ dom A C_ B ) -> A C_ ( A |` B ) )
12 resss 
 |-  ( A |` B ) C_ A
13 11 12 jctil 
 |-  ( ( Rel A /\ dom A C_ B ) -> ( ( A |` B ) C_ A /\ A C_ ( A |` B ) ) )
14 eqss 
 |-  ( ( A |` B ) = A <-> ( ( A |` B ) C_ A /\ A C_ ( A |` B ) ) )
15 13 14 sylibr 
 |-  ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A )