Metamath Proof Explorer


Theorem dffunsALTV2

Description: Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021)

Ref Expression
Assertion dffunsALTV2 FunsALTV = { 𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I }

Proof

Step Hyp Ref Expression
1 dffunsALTV FunsALTV = { 𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
2 cosselcnvrefrels2 ( ≀ 𝑓 ∈ CnvRefRels ↔ ( ≀ 𝑓 ⊆ I ∧ ≀ 𝑓 ∈ Rels ) )
3 cosselrels ( 𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels )
4 3 biantrud ( 𝑓 ∈ Rels → ( ≀ 𝑓 ⊆ I ↔ ( ≀ 𝑓 ⊆ I ∧ ≀ 𝑓 ∈ Rels ) ) )
5 2 4 bitr4id ( 𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ≀ 𝑓 ⊆ I ) )
6 1 5 rabimbieq FunsALTV = { 𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I }