Step |
Hyp |
Ref |
Expression |
1 |
|
cossex |
⊢ ( 𝐹 ∈ 𝑉 → ≀ 𝐹 ∈ V ) |
2 |
|
elcnvrefrelsrel |
⊢ ( ≀ 𝐹 ∈ V → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹 ) ) |
3 |
1 2
|
syl |
⊢ ( 𝐹 ∈ 𝑉 → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹 ) ) |
4 |
|
elrelsrel |
⊢ ( 𝐹 ∈ 𝑉 → ( 𝐹 ∈ Rels ↔ Rel 𝐹 ) ) |
5 |
3 4
|
anbi12d |
⊢ ( 𝐹 ∈ 𝑉 → ( ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹 ) ) ) |
6 |
|
elfunsALTV |
⊢ ( 𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ) |
7 |
|
df-funALTV |
⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹 ) ) |
8 |
5 6 7
|
3bitr4g |
⊢ ( 𝐹 ∈ 𝑉 → ( 𝐹 ∈ FunsALTV ↔ FunALTV 𝐹 ) ) |