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Metamath Proof Explorer

Theorem elopabi

Description: A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006)

Ref Expression
Hypotheses elopabi.1 
|- ( x = ( 1st ` A ) -> ( ph <-> ps ) )
elopabi.2 
|- ( y = ( 2nd ` A ) -> ( ps <-> ch ) )
Assertion elopabi 
|- ( A e. { <. x , y >. | ph } -> ch )

Proof

Step Hyp Ref Expression
1 elopabi.1 
 |-  ( x = ( 1st ` A ) -> ( ph <-> ps ) )
2 elopabi.2 
 |-  ( y = ( 2nd ` A ) -> ( ps <-> ch ) )
3 relopabv 
 |-  Rel { <. x , y >. | ph }
4 1st2nd 
 |-  ( ( Rel { <. x , y >. | ph } /\ A e. { <. x , y >. | ph } ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
5 3 4 mpan 
 |-  ( A e. { <. x , y >. | ph } -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
6 id 
 |-  ( A e. { <. x , y >. | ph } -> A e. { <. x , y >. | ph } )
7 5 6 eqeltrrd 
 |-  ( A e. { <. x , y >. | ph } -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } )
8 fvex 
 |-  ( 1st ` A ) e. _V
9 fvex 
 |-  ( 2nd ` A ) e. _V
10 8 9 1 2 opelopab 
 |-  ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } <-> ch )
11 7 10 sylib 
 |-  ( A e. { <. x , y >. | ph } -> ch )