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Metamath Proof Explorer

Theorem tz6.12

Description: Function value. Theorem 6.12(1) of TakeutiZaring p. 27. (Contributed by NM, 10-Jul-1994)

Ref Expression
Assertion tz6.12 ( ( ⟨ 𝐴 , 𝑦 ⟩ ∈ 𝐹 ∧ ∃! 𝑦 ⟨ 𝐴 , 𝑦 ⟩ ∈ 𝐹 ) → ( 𝐹 ‘ 𝐴 ) = 𝑦 )

Proof

Step Hyp Ref Expression
1 df-br ⊢ ( 𝐴 𝐹 𝑦 ↔ ⟨ 𝐴 , 𝑦 ⟩ ∈ 𝐹 )
2 1 eubii ⊢ ( ∃! 𝑦 𝐴 𝐹 𝑦 ↔ ∃! 𝑦 ⟨ 𝐴 , 𝑦 ⟩ ∈ 𝐹 )
3 tz6.12-1 ⊢ ( ( 𝐴 𝐹 𝑦 ∧ ∃! 𝑦 𝐴 𝐹 𝑦 ) → ( 𝐹 ‘ 𝐴 ) = 𝑦 )
4 1 2 3 syl2anbr ⊢ ( ( ⟨ 𝐴 , 𝑦 ⟩ ∈ 𝐹 ∧ ∃! 𝑦 ⟨ 𝐴 , 𝑦 ⟩ ∈ 𝐹 ) → ( 𝐹 ‘ 𝐴 ) = 𝑦 )