Metamath Proof Explorer
Description: A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011)
|
|
Ref |
Expression |
|
Assertion |
resdisj |
⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝐶 ↾ 𝐴 ) ↾ 𝐵 ) = ∅ ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
reseq2 |
⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝐶 ↾ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐶 ↾ ∅ ) ) |
2 |
|
resres |
⊢ ( ( 𝐶 ↾ 𝐴 ) ↾ 𝐵 ) = ( 𝐶 ↾ ( 𝐴 ∩ 𝐵 ) ) |
3 |
|
res0 |
⊢ ( 𝐶 ↾ ∅ ) = ∅ |
4 |
3
|
eqcomi |
⊢ ∅ = ( 𝐶 ↾ ∅ ) |
5 |
1 2 4
|
3eqtr4g |
⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝐶 ↾ 𝐴 ) ↾ 𝐵 ) = ∅ ) |