Step |
Hyp |
Ref |
Expression |
1 |
|
df-refrel |
⊢ ( RefRel 𝑅 ↔ ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) ) |
2 |
|
dfrel6 |
⊢ ( Rel 𝑅 ↔ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) = 𝑅 ) |
3 |
2
|
biimpi |
⊢ ( Rel 𝑅 → ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) = 𝑅 ) |
4 |
3
|
sseq2d |
⊢ ( Rel 𝑅 → ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ↔ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ) ) |
5 |
4
|
pm5.32ri |
⊢ ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) ↔ ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) ) |
6 |
1 5
|
bitri |
⊢ ( RefRel 𝑅 ↔ ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) ) |