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Metamath Proof Explorer

Theorem dfrab3ss

Description: Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015) (Proof shortened by Mario Carneiro, 8-Nov-2015)

Ref Expression
Assertion dfrab3ss 
|- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) )

Proof

Step Hyp Ref Expression
1 dfss2 
 |-  ( A C_ B <-> ( A i^i B ) = A )
2 ineq1 
 |-  ( ( A i^i B ) = A -> ( ( A i^i B ) i^i { x | ph } ) = ( A i^i { x | ph } ) )
3 2 eqcomd 
 |-  ( ( A i^i B ) = A -> ( A i^i { x | ph } ) = ( ( A i^i B ) i^i { x | ph } ) )
4 1 3 sylbi 
 |-  ( A C_ B -> ( A i^i { x | ph } ) = ( ( A i^i B ) i^i { x | ph } ) )
5 dfrab3 
 |-  { x e. A | ph } = ( A i^i { x | ph } )
6 dfrab3 
 |-  { x e. B | ph } = ( B i^i { x | ph } )
7 6 ineq2i 
 |-  ( A i^i { x e. B | ph } ) = ( A i^i ( B i^i { x | ph } ) )
8 inass 
 |-  ( ( A i^i B ) i^i { x | ph } ) = ( A i^i ( B i^i { x | ph } ) )
9 7 8 eqtr4i 
 |-  ( A i^i { x e. B | ph } ) = ( ( A i^i B ) i^i { x | ph } )
10 4 5 9 3eqtr4g 
 |-  ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) )