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Metamath Proof Explorer
Description: Two things in a binary relation belong to the relation's domain.
(Contributed by NM, 17-May-1996) (Revised by Mario Carneiro, 26-Apr-2015)
|
|
Ref |
Expression |
|
Hypothesis |
brel.1 |
⊢ 𝑅 ⊆ ( 𝐶 × 𝐷 ) |
|
Assertion |
brel |
⊢ ( 𝐴 𝑅 𝐵 → ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
brel.1 |
⊢ 𝑅 ⊆ ( 𝐶 × 𝐷 ) |
| 2 |
1
|
ssbri |
⊢ ( 𝐴 𝑅 𝐵 → 𝐴 ( 𝐶 × 𝐷 ) 𝐵 ) |
| 3 |
|
brxp |
⊢ ( 𝐴 ( 𝐶 × 𝐷 ) 𝐵 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ) |
| 4 |
2 3
|
sylib |
⊢ ( 𝐴 𝑅 𝐵 → ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ) |