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Metamath Proof Explorer

Theorem resindi

Description: Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008)

Ref Expression
Assertion resindi 
|- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) )

Proof

Step Hyp Ref Expression
1 xpindir 
 |-  ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
2 1 ineq2i 
 |-  ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
3 inindi 
 |-  ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) ) = ( ( A i^i ( B X. _V ) ) i^i ( A i^i ( C X. _V ) ) )
4 2 3 eqtri 
 |-  ( A i^i ( ( B i^i C ) X. _V ) ) = ( ( A i^i ( B X. _V ) ) i^i ( A i^i ( C X. _V ) ) )
5 df-res 
 |-  ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
6 df-res 
 |-  ( A |` B ) = ( A i^i ( B X. _V ) )
7 df-res 
 |-  ( A |` C ) = ( A i^i ( C X. _V ) )
8 6 7 ineq12i 
 |-  ( ( A |` B ) i^i ( A |` C ) ) = ( ( A i^i ( B X. _V ) ) i^i ( A i^i ( C X. _V ) ) )
9 4 5 8 3eqtr4i 
 |-  ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) )