Metamath Proof Explorer


Theorem resdifdi

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

Ref Expression
Assertion resdifdi ( 𝐴 ↾ ( 𝐵𝐶 ) ) = ( ( 𝐴𝐵 ) ∖ ( 𝐴𝐶 ) )

Proof

Step Hyp Ref Expression
1 df-res ( 𝐴 ↾ ( 𝐵𝐶 ) ) = ( 𝐴 ∩ ( ( 𝐵𝐶 ) × V ) )
2 difxp1 ( ( 𝐵𝐶 ) × V ) = ( ( 𝐵 × V ) ∖ ( 𝐶 × V ) )
3 2 ineq2i ( 𝐴 ∩ ( ( 𝐵𝐶 ) × V ) ) = ( 𝐴 ∩ ( ( 𝐵 × V ) ∖ ( 𝐶 × V ) ) )
4 indifdi ( 𝐴 ∩ ( ( 𝐵 × V ) ∖ ( 𝐶 × V ) ) ) = ( ( 𝐴 ∩ ( 𝐵 × V ) ) ∖ ( 𝐴 ∩ ( 𝐶 × V ) ) )
5 1 3 4 3eqtri ( 𝐴 ↾ ( 𝐵𝐶 ) ) = ( ( 𝐴 ∩ ( 𝐵 × V ) ) ∖ ( 𝐴 ∩ ( 𝐶 × V ) ) )
6 df-res ( 𝐴𝐵 ) = ( 𝐴 ∩ ( 𝐵 × V ) )
7 df-res ( 𝐴𝐶 ) = ( 𝐴 ∩ ( 𝐶 × V ) )
8 6 7 difeq12i ( ( 𝐴𝐵 ) ∖ ( 𝐴𝐶 ) ) = ( ( 𝐴 ∩ ( 𝐵 × V ) ) ∖ ( 𝐴 ∩ ( 𝐶 × V ) ) )
9 5 8 eqtr4i ( 𝐴 ↾ ( 𝐵𝐶 ) ) = ( ( 𝐴𝐵 ) ∖ ( 𝐴𝐶 ) )