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Metamath Proof Explorer

Theorem resres

Description: The restriction of a restriction. (Contributed by NM, 27-Mar-2008)

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

Proof

Step Hyp Ref Expression
1 df-res 
 |-  ( ( A |` B ) |` C ) = ( ( A |` B ) i^i ( C X. _V ) )
2 df-res 
 |-  ( A |` B ) = ( A i^i ( B X. _V ) )
3 2 ineq1i 
 |-  ( ( A |` B ) i^i ( C X. _V ) ) = ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) )
4 xpindir 
 |-  ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
5 4 ineq2i 
 |-  ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
6 df-res 
 |-  ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
7 inass 
 |-  ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
8 5 6 7 3eqtr4ri 
 |-  ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) ) = ( A |` ( B i^i C ) )
9 1 3 8 3eqtri 
 |-  ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )