Description: Move intersection into class restriction. (Contributed by NM, 18-Dec-2008)
Ref | Expression | ||
---|---|---|---|
Assertion | inres | ⊢ ( 𝐴 ∩ ( 𝐵 ↾ 𝐶 ) ) = ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass | ⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐶 × V ) ) = ( 𝐴 ∩ ( 𝐵 ∩ ( 𝐶 × V ) ) ) | |
2 | df-res | ⊢ ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐶 × V ) ) | |
3 | df-res | ⊢ ( 𝐵 ↾ 𝐶 ) = ( 𝐵 ∩ ( 𝐶 × V ) ) | |
4 | 3 | ineq2i | ⊢ ( 𝐴 ∩ ( 𝐵 ↾ 𝐶 ) ) = ( 𝐴 ∩ ( 𝐵 ∩ ( 𝐶 × V ) ) ) |
5 | 1 2 4 | 3eqtr4ri | ⊢ ( 𝐴 ∩ ( 𝐵 ↾ 𝐶 ) ) = ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 ) |