Metamath Proof Explorer

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