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Metamath Proof Explorer

Theorem dffun4

Description: Alternate definition of a function. Definition 6.4(4) of TakeutiZaring p. 24. (Contributed by NM, 29-Dec-1996)

Ref Expression
Assertion dffun4 ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐴 ) → 𝑦 = 𝑧 ) ) )

Proof

Step Hyp Ref Expression
1 dffun2 ⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ) )
2 df-br ⊢ ( 𝑥 𝐴 𝑦 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 )
3 df-br ⊢ ( 𝑥 𝐴 𝑧 ↔ ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐴 )
4 2 3 anbi12i ⊢ ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) ↔ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐴 ) )
5 4 imbi1i ⊢ ( ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ↔ ( ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐴 ) → 𝑦 = 𝑧 ) )
6 5 albii ⊢ ( ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ↔ ∀ 𝑧 ( ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐴 ) → 𝑦 = 𝑧 ) )
7 6 2albii ⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐴 ) → 𝑦 = 𝑧 ) )
8 7 anbi2i ⊢ ( ( Rel 𝐴 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ) ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐴 ) → 𝑦 = 𝑧 ) ) )
9 1 8 bitri ⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐴 ) → 𝑦 = 𝑧 ) ) )