Metamath Proof Explorer

Theorem rabss2

Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004) (Proof shortened by Andrew Salmon, 26-Jun-2011)

Ref Expression
Assertion rabss2 
|- ( A C_ B -> { x e. A | ph } C_ { x e. B | ph } )

Proof

Step Hyp Ref Expression
1 pm3.45 
 |-  ( ( x e. A -> x e. B ) -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
2 1 alimi 
 |-  ( A. x ( x e. A -> x e. B ) -> A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
3 dfss2 
 |-  ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4 ss2ab 
 |-  ( { x | ( x e. A /\ ph ) } C_ { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
5 2 3 4 3imtr4i 
 |-  ( A C_ B -> { x | ( x e. A /\ ph ) } C_ { x | ( x e. B /\ ph ) } )
6 df-rab 
 |-  { x e. A | ph } = { x | ( x e. A /\ ph ) }
7 df-rab 
 |-  { x e. B | ph } = { x | ( x e. B /\ ph ) }
8 5 6 7 3sstr4g 
 |-  ( A C_ B -> { x e. A | ph } C_ { x e. B | ph } )