Theorem dffunsALTV3

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 A. y1 A. y2 ( ( x1 f y1 /\ x1 f y2 ) -> y1 = y2 ) } . (Contributed by Peter Mazsa, 30-Aug-2021)

	Ref	Expression
Assertion	dffunsALTV3	⊢ FunsALTV = { 𝑓 ∈ Rels ∣ ∀ 𝑢 ∀ 𝑥 ∀ 𝑦 ( ( 𝑢 𝑓 𝑥 ∧ 𝑢 𝑓 𝑦 ) → 𝑥 = 𝑦 ) }

Proof

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 ) }