Step |
Hyp |
Ref |
Expression |
1 |
|
df-coss |
⊢ ≀ 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑅 𝑦 ) } |
2 |
|
elecALTV |
⊢ ( ( 𝑢 ∈ V ∧ 𝑥 ∈ V ) → ( 𝑥 ∈ [ 𝑢 ] 𝑅 ↔ 𝑢 𝑅 𝑥 ) ) |
3 |
2
|
el2v |
⊢ ( 𝑥 ∈ [ 𝑢 ] 𝑅 ↔ 𝑢 𝑅 𝑥 ) |
4 |
|
elecALTV |
⊢ ( ( 𝑢 ∈ V ∧ 𝑦 ∈ V ) → ( 𝑦 ∈ [ 𝑢 ] 𝑅 ↔ 𝑢 𝑅 𝑦 ) ) |
5 |
4
|
el2v |
⊢ ( 𝑦 ∈ [ 𝑢 ] 𝑅 ↔ 𝑢 𝑅 𝑦 ) |
6 |
3 5
|
anbi12i |
⊢ ( ( 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝑦 ∈ [ 𝑢 ] 𝑅 ) ↔ ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑅 𝑦 ) ) |
7 |
6
|
exbii |
⊢ ( ∃ 𝑢 ( 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝑦 ∈ [ 𝑢 ] 𝑅 ) ↔ ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑅 𝑦 ) ) |
8 |
7
|
opabbii |
⊢ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ( 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝑦 ∈ [ 𝑢 ] 𝑅 ) } = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑅 𝑦 ) } |
9 |
1 8
|
eqtr4i |
⊢ ≀ 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ( 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝑦 ∈ [ 𝑢 ] 𝑅 ) } |