Metamath Proof Explorer
Description: The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013)
|
|
Ref |
Expression |
|
Hypotheses |
opelopaba.1 |
⊢ 𝐴 ∈ V |
|
|
opelopaba.2 |
⊢ 𝐵 ∈ V |
|
|
opelopaba.3 |
⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) |
|
|
braba.4 |
⊢ 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ 𝜑 } |
|
Assertion |
braba |
⊢ ( 𝐴 𝑅 𝐵 ↔ 𝜓 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
opelopaba.1 |
⊢ 𝐴 ∈ V |
2 |
|
opelopaba.2 |
⊢ 𝐵 ∈ V |
3 |
|
opelopaba.3 |
⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) |
4 |
|
braba.4 |
⊢ 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ 𝜑 } |
5 |
3 4
|
brabga |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 𝑅 𝐵 ↔ 𝜓 ) ) |
6 |
1 2 5
|
mp2an |
⊢ ( 𝐴 𝑅 𝐵 ↔ 𝜓 ) |