Metamath Proof Explorer

Theorem setsval

Description: Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014) (Revised by Mario Carneiro, 30-Apr-2015)

Ref Expression
Assertion setsval 
|- ( ( S e. V /\ B e. W ) -> ( S sSet <. A , B >. ) = ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) )

Proof

Step Hyp Ref Expression
1 opex 
 |-  <. A , B >. e. _V
2 setsvalg 
 |-  ( ( S e. V /\ <. A , B >. e. _V ) -> ( S sSet <. A , B >. ) = ( ( S |` ( _V \ dom { <. A , B >. } ) ) u. { <. A , B >. } ) )
3 1 2 mpan2 
 |-  ( S e. V -> ( S sSet <. A , B >. ) = ( ( S |` ( _V \ dom { <. A , B >. } ) ) u. { <. A , B >. } ) )
4 dmsnopg 
 |-  ( B e. W -> dom { <. A , B >. } = { A } )
5 4 difeq2d 
 |-  ( B e. W -> ( _V \ dom { <. A , B >. } ) = ( _V \ { A } ) )
6 5 reseq2d 
 |-  ( B e. W -> ( S |` ( _V \ dom { <. A , B >. } ) ) = ( S |` ( _V \ { A } ) ) )
7 6 uneq1d 
 |-  ( B e. W -> ( ( S |` ( _V \ dom { <. A , B >. } ) ) u. { <. A , B >. } ) = ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) )
8 3 7 sylan9eq 
 |-  ( ( S e. V /\ B e. W ) -> ( S sSet <. A , B >. ) = ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) )