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