Metamath Proof Explorer


Theorem elfunsALTV3

Description: Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021)

Ref Expression
Assertion elfunsALTV3 ( 𝐹 ∈ FunsALTV ↔ ( ∀ 𝑢𝑥𝑦 ( ( 𝑢 𝐹 𝑥𝑢 𝐹 𝑦 ) → 𝑥 = 𝑦 ) ∧ 𝐹 ∈ Rels ) )

Proof

Step Hyp Ref Expression
1 elfunsALTV ( 𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) )
2 cosselcnvrefrels3 ( ≀ 𝐹 ∈ CnvRefRels ↔ ( ∀ 𝑢𝑥𝑦 ( ( 𝑢 𝐹 𝑥𝑢 𝐹 𝑦 ) → 𝑥 = 𝑦 ) ∧ ≀ 𝐹 ∈ Rels ) )
3 cosselrels ( 𝐹 ∈ Rels → ≀ 𝐹 ∈ Rels )
4 3 biantrud ( 𝐹 ∈ Rels → ( ∀ 𝑢𝑥𝑦 ( ( 𝑢 𝐹 𝑥𝑢 𝐹 𝑦 ) → 𝑥 = 𝑦 ) ↔ ( ∀ 𝑢𝑥𝑦 ( ( 𝑢 𝐹 𝑥𝑢 𝐹 𝑦 ) → 𝑥 = 𝑦 ) ∧ ≀ 𝐹 ∈ Rels ) ) )
5 2 4 bitr4id ( 𝐹 ∈ Rels → ( ≀ 𝐹 ∈ CnvRefRels ↔ ∀ 𝑢𝑥𝑦 ( ( 𝑢 𝐹 𝑥𝑢 𝐹 𝑦 ) → 𝑥 = 𝑦 ) ) )
6 5 pm5.32ri ( ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ ( ∀ 𝑢𝑥𝑦 ( ( 𝑢 𝐹 𝑥𝑢 𝐹 𝑦 ) → 𝑥 = 𝑦 ) ∧ 𝐹 ∈ Rels ) )
7 1 6 bitri ( 𝐹 ∈ FunsALTV ↔ ( ∀ 𝑢𝑥𝑦 ( ( 𝑢 𝐹 𝑥𝑢 𝐹 𝑦 ) → 𝑥 = 𝑦 ) ∧ 𝐹 ∈ Rels ) )