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 }