Theorem dffunsALTV4

Description: Alternate definition of the class of functions. For the X axis and the Y axis you can convert the right side to { f e. Rels | A. x1 E* y1 x1 f y1 } . (Contributed by Peter Mazsa, 31-Aug-2021)

	Ref	Expression
Assertion	dffunsALTV4	⊢ FunsALTV = { 𝑓 ∈ Rels ∣ ∀ 𝑢 ∃* 𝑥 𝑢 𝑓 𝑥 }

Proof

Step	Hyp	Ref	Expression
1		dffunsALTV	⊢ FunsALTV = { 𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
2		cosselcnvrefrels4	⊢ ( ≀ 𝑓 ∈ CnvRefRels ↔ ( ∀ 𝑢 ∃* 𝑥 𝑢 𝑓 𝑥 ∧ ≀ 𝑓 ∈ Rels ) )
3		cosselrels	⊢ ( 𝑓 ∈ Rels → ≀ 𝑓 ∈ Rels )
4	3	biantrud	⊢ ( 𝑓 ∈ Rels → ( ∀ 𝑢 ∃* 𝑥 𝑢 𝑓 𝑥 ↔ ( ∀ 𝑢 ∃* 𝑥 𝑢 𝑓 𝑥 ∧ ≀ 𝑓 ∈ Rels ) ) )
5	2 4	bitr4id	⊢ ( 𝑓 ∈ Rels → ( ≀ 𝑓 ∈ CnvRefRels ↔ ∀ 𝑢 ∃* 𝑥 𝑢 𝑓 𝑥 ) )
6	1 5	rabimbieq	⊢ FunsALTV = { 𝑓 ∈ Rels ∣ ∀ 𝑢 ∃* 𝑥 𝑢 𝑓 𝑥 }