Step |
Hyp |
Ref |
Expression |
1 |
|
isdir.1 |
⊢ 𝐴 = ∪ ∪ 𝑅 |
2 |
|
releq |
⊢ ( 𝑟 = 𝑅 → ( Rel 𝑟 ↔ Rel 𝑅 ) ) |
3 |
|
unieq |
⊢ ( 𝑟 = 𝑅 → ∪ 𝑟 = ∪ 𝑅 ) |
4 |
3
|
unieqd |
⊢ ( 𝑟 = 𝑅 → ∪ ∪ 𝑟 = ∪ ∪ 𝑅 ) |
5 |
4 1
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ∪ ∪ 𝑟 = 𝐴 ) |
6 |
5
|
reseq2d |
⊢ ( 𝑟 = 𝑅 → ( I ↾ ∪ ∪ 𝑟 ) = ( I ↾ 𝐴 ) ) |
7 |
|
id |
⊢ ( 𝑟 = 𝑅 → 𝑟 = 𝑅 ) |
8 |
6 7
|
sseq12d |
⊢ ( 𝑟 = 𝑅 → ( ( I ↾ ∪ ∪ 𝑟 ) ⊆ 𝑟 ↔ ( I ↾ 𝐴 ) ⊆ 𝑅 ) ) |
9 |
2 8
|
anbi12d |
⊢ ( 𝑟 = 𝑅 → ( ( Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟 ) ⊆ 𝑟 ) ↔ ( Rel 𝑅 ∧ ( I ↾ 𝐴 ) ⊆ 𝑅 ) ) ) |
10 |
7 7
|
coeq12d |
⊢ ( 𝑟 = 𝑅 → ( 𝑟 ∘ 𝑟 ) = ( 𝑅 ∘ 𝑅 ) ) |
11 |
10 7
|
sseq12d |
⊢ ( 𝑟 = 𝑅 → ( ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ↔ ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ) ) |
12 |
5
|
sqxpeqd |
⊢ ( 𝑟 = 𝑅 → ( ∪ ∪ 𝑟 × ∪ ∪ 𝑟 ) = ( 𝐴 × 𝐴 ) ) |
13 |
|
cnveq |
⊢ ( 𝑟 = 𝑅 → ◡ 𝑟 = ◡ 𝑅 ) |
14 |
13 7
|
coeq12d |
⊢ ( 𝑟 = 𝑅 → ( ◡ 𝑟 ∘ 𝑟 ) = ( ◡ 𝑅 ∘ 𝑅 ) ) |
15 |
12 14
|
sseq12d |
⊢ ( 𝑟 = 𝑅 → ( ( ∪ ∪ 𝑟 × ∪ ∪ 𝑟 ) ⊆ ( ◡ 𝑟 ∘ 𝑟 ) ↔ ( 𝐴 × 𝐴 ) ⊆ ( ◡ 𝑅 ∘ 𝑅 ) ) ) |
16 |
11 15
|
anbi12d |
⊢ ( 𝑟 = 𝑅 → ( ( ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ∧ ( ∪ ∪ 𝑟 × ∪ ∪ 𝑟 ) ⊆ ( ◡ 𝑟 ∘ 𝑟 ) ) ↔ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ ( 𝐴 × 𝐴 ) ⊆ ( ◡ 𝑅 ∘ 𝑅 ) ) ) ) |
17 |
9 16
|
anbi12d |
⊢ ( 𝑟 = 𝑅 → ( ( ( Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟 ) ⊆ 𝑟 ) ∧ ( ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ∧ ( ∪ ∪ 𝑟 × ∪ ∪ 𝑟 ) ⊆ ( ◡ 𝑟 ∘ 𝑟 ) ) ) ↔ ( ( Rel 𝑅 ∧ ( I ↾ 𝐴 ) ⊆ 𝑅 ) ∧ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ ( 𝐴 × 𝐴 ) ⊆ ( ◡ 𝑅 ∘ 𝑅 ) ) ) ) ) |
18 |
|
df-dir |
⊢ DirRel = { 𝑟 ∣ ( ( Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟 ) ⊆ 𝑟 ) ∧ ( ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ∧ ( ∪ ∪ 𝑟 × ∪ ∪ 𝑟 ) ⊆ ( ◡ 𝑟 ∘ 𝑟 ) ) ) } |
19 |
17 18
|
elab2g |
⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ DirRel ↔ ( ( Rel 𝑅 ∧ ( I ↾ 𝐴 ) ⊆ 𝑅 ) ∧ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ ( 𝐴 × 𝐴 ) ⊆ ( ◡ 𝑅 ∘ 𝑅 ) ) ) ) ) |