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Metamath Proof Explorer

Theorem resdifdi

Description: Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024)

Ref Expression
Assertion resdifdi 
|- ( A |` ( B \ C ) ) = ( ( A |` B ) \ ( A |` C ) )

Proof

Step Hyp Ref Expression
1 df-res 
 |-  ( A |` ( B \ C ) ) = ( A i^i ( ( B \ C ) X. _V ) )
2 difxp1 
 |-  ( ( B \ C ) X. _V ) = ( ( B X. _V ) \ ( C X. _V ) )
3 2 ineq2i 
 |-  ( A i^i ( ( B \ C ) X. _V ) ) = ( A i^i ( ( B X. _V ) \ ( C X. _V ) ) )
4 indifdi 
 |-  ( A i^i ( ( B X. _V ) \ ( C X. _V ) ) ) = ( ( A i^i ( B X. _V ) ) \ ( A i^i ( C X. _V ) ) )
5 1 3 4 3eqtri 
 |-  ( A |` ( B \ C ) ) = ( ( A i^i ( B X. _V ) ) \ ( A 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 difeq12i 
 |-  ( ( A |` B ) \ ( A |` C ) ) = ( ( A i^i ( B X. _V ) ) \ ( A i^i ( C X. _V ) ) )
9 5 8 eqtr4i 
 |-  ( A |` ( B \ C ) ) = ( ( A |` B ) \ ( A |` C ) )