Step |
Hyp |
Ref |
Expression |
1 |
|
dfeqvrels2 |
⊢ EqvRels = { 𝑟 ∈ Rels ∣ ( ( I ↾ dom 𝑟 ) ⊆ 𝑟 ∧ ◡ 𝑟 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } |
2 |
|
idrefALT |
⊢ ( ( I ↾ dom 𝑟 ) ⊆ 𝑟 ↔ ∀ 𝑥 ∈ dom 𝑟 𝑥 𝑟 𝑥 ) |
3 |
|
cnvsym |
⊢ ( ◡ 𝑟 ⊆ 𝑟 ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑟 𝑦 → 𝑦 𝑟 𝑥 ) ) |
4 |
|
cotr |
⊢ ( ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) → 𝑥 𝑟 𝑧 ) ) |
5 |
2 3 4
|
3anbi123i |
⊢ ( ( ( I ↾ dom 𝑟 ) ⊆ 𝑟 ∧ ◡ 𝑟 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) ↔ ( ∀ 𝑥 ∈ dom 𝑟 𝑥 𝑟 𝑥 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑟 𝑦 → 𝑦 𝑟 𝑥 ) ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) → 𝑥 𝑟 𝑧 ) ) ) |
6 |
1 5
|
rabbieq |
⊢ EqvRels = { 𝑟 ∈ Rels ∣ ( ∀ 𝑥 ∈ dom 𝑟 𝑥 𝑟 𝑥 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑟 𝑦 → 𝑦 𝑟 𝑥 ) ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝑟 𝑦 ∧ 𝑦 𝑟 𝑧 ) → 𝑥 𝑟 𝑧 ) ) } |