Metamath Proof Explorer


Theorem dffunALTV5

Description: Alternate definition of the function relation predicate, cf. dfdisjALTV5 . (Contributed by Peter Mazsa, 5-Sep-2021)

Ref Expression
Assertion dffunALTV5 ( FunALTV 𝐹 ↔ ( ∀ 𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] 𝐹 ∩ [ 𝑦 ] 𝐹 ) = ∅ ) ∧ Rel 𝐹 ) )

Proof

Step Hyp Ref Expression
1 dffunALTV2 ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹 ) )
2 cossssid5 ( ≀ 𝐹 ⊆ I ↔ ∀ 𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] 𝐹 ∩ [ 𝑦 ] 𝐹 ) = ∅ ) )
3 2 anbi1i ( ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹 ) ↔ ( ∀ 𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] 𝐹 ∩ [ 𝑦 ] 𝐹 ) = ∅ ) ∧ Rel 𝐹 ) )
4 1 3 bitri ( FunALTV 𝐹 ↔ ( ∀ 𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] 𝐹 ∩ [ 𝑦 ] 𝐹 ) = ∅ ) ∧ Rel 𝐹 ) )