Step |
Hyp |
Ref |
Expression |
1 |
|
eldmqsres |
⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ ( dom ( 𝑅 ↾ 𝐴 ) / ( 𝑅 ↾ 𝐴 ) ) ↔ ∃ 𝑢 ∈ 𝐴 ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) ) |
2 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐵 = [ 𝑢 ] 𝑅 ↔ ∃ 𝑥 ( 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) |
3 |
|
19.41v |
⊢ ( ∃ 𝑥 ( 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ↔ ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) |
4 |
2 3
|
bitri |
⊢ ( ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐵 = [ 𝑢 ] 𝑅 ↔ ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) |
5 |
4
|
rexbii |
⊢ ( ∃ 𝑢 ∈ 𝐴 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐵 = [ 𝑢 ] 𝑅 ↔ ∃ 𝑢 ∈ 𝐴 ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) |
6 |
1 5
|
bitr4di |
⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ ( dom ( 𝑅 ↾ 𝐴 ) / ( 𝑅 ↾ 𝐴 ) ) ↔ ∃ 𝑢 ∈ 𝐴 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐵 = [ 𝑢 ] 𝑅 ) ) |