Step |
Hyp |
Ref |
Expression |
1 |
|
df-res |
⊢ ( 𝐴 ↾ ( 𝐵 ∖ 𝐶 ) ) = ( 𝐴 ∩ ( ( 𝐵 ∖ 𝐶 ) × V ) ) |
2 |
|
difxp1 |
⊢ ( ( 𝐵 ∖ 𝐶 ) × V ) = ( ( 𝐵 × V ) ∖ ( 𝐶 × V ) ) |
3 |
2
|
ineq2i |
⊢ ( 𝐴 ∩ ( ( 𝐵 ∖ 𝐶 ) × 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 ↾ ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 ∩ ( 𝐵 × V ) ) ∖ ( 𝐴 ∩ ( 𝐶 × V ) ) ) |
6 |
|
df-res |
⊢ ( 𝐴 ↾ 𝐵 ) = ( 𝐴 ∩ ( 𝐵 × V ) ) |
7 |
|
df-res |
⊢ ( 𝐴 ↾ 𝐶 ) = ( 𝐴 ∩ ( 𝐶 × V ) ) |
8 |
6 7
|
difeq12i |
⊢ ( ( 𝐴 ↾ 𝐵 ) ∖ ( 𝐴 ↾ 𝐶 ) ) = ( ( 𝐴 ∩ ( 𝐵 × V ) ) ∖ ( 𝐴 ∩ ( 𝐶 × V ) ) ) |
9 |
5 8
|
eqtr4i |
⊢ ( 𝐴 ↾ ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 ↾ 𝐵 ) ∖ ( 𝐴 ↾ 𝐶 ) ) |