Step |
Hyp |
Ref |
Expression |
1 |
|
dffunsALTV |
⊢ FunsALTV = { 𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels } |
2 |
|
cosselcnvrefrels3 |
⊢ ( ≀ 𝑓 ∈ CnvRefRels ↔ ( ∀ 𝑢 ∀ 𝑥 ∀ 𝑦 ( ( 𝑢 𝑓 𝑥 ∧ 𝑢 𝑓 𝑦 ) → 𝑥 = 𝑦 ) ∧ ≀ 𝑓 ∈ Rels ) ) |
3 |
|
cosselrels |
⊢ ( 𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels ) |
4 |
3
|
biantrud |
⊢ ( 𝑓 ∈ Rels → ( ∀ 𝑢 ∀ 𝑥 ∀ 𝑦 ( ( 𝑢 𝑓 𝑥 ∧ 𝑢 𝑓 𝑦 ) → 𝑥 = 𝑦 ) ↔ ( ∀ 𝑢 ∀ 𝑥 ∀ 𝑦 ( ( 𝑢 𝑓 𝑥 ∧ 𝑢 𝑓 𝑦 ) → 𝑥 = 𝑦 ) ∧ ≀ 𝑓 ∈ Rels ) ) ) |
5 |
2 4
|
bitr4id |
⊢ ( 𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ∀ 𝑢 ∀ 𝑥 ∀ 𝑦 ( ( 𝑢 𝑓 𝑥 ∧ 𝑢 𝑓 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
6 |
1 5
|
rabimbieq |
⊢ FunsALTV = { 𝑓 ∈ Rels ∣ ∀ 𝑢 ∀ 𝑥 ∀ 𝑦 ( ( 𝑢 𝑓 𝑥 ∧ 𝑢 𝑓 𝑦 ) → 𝑥 = 𝑦 ) } |