Step |
Hyp |
Ref |
Expression |
1 |
|
elfunsALTV |
⊢ ( 𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ) |
2 |
|
cosselcnvrefrels5 |
⊢ ( ≀ 𝐹 ∈ CnvRefRels ↔ ( ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝐹 ∩ [ 𝑦 ] ◡ 𝐹 ) = ∅ ) ∧ ≀ 𝐹 ∈ Rels ) ) |
3 |
|
cosselrels |
⊢ ( 𝐹 ∈ Rels → ≀ 𝐹 ∈ Rels ) |
4 |
3
|
biantrud |
⊢ ( 𝐹 ∈ Rels → ( ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝐹 ∩ [ 𝑦 ] ◡ 𝐹 ) = ∅ ) ↔ ( ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝐹 ∩ [ 𝑦 ] ◡ 𝐹 ) = ∅ ) ∧ ≀ 𝐹 ∈ Rels ) ) ) |
5 |
2 4
|
bitr4id |
⊢ ( 𝐹 ∈ Rels → ( ≀ 𝐹 ∈ CnvRefRels ↔ ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝐹 ∩ [ 𝑦 ] ◡ 𝐹 ) = ∅ ) ) ) |
6 |
5
|
pm5.32ri |
⊢ ( ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ ( ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝐹 ∩ [ 𝑦 ] ◡ 𝐹 ) = ∅ ) ∧ 𝐹 ∈ Rels ) ) |
7 |
1 6
|
bitri |
⊢ ( 𝐹 ∈ FunsALTV ↔ ( ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝐹 ∩ [ 𝑦 ] ◡ 𝐹 ) = ∅ ) ∧ 𝐹 ∈ Rels ) ) |