Metamath Proof Explorer


Theorem tz6.12-1OLD

Description: Obsolete version of tz6.12-1 as of 23-Dec-2024. (Contributed by NM, 30-Apr-2004) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion tz6.12-1OLD ( ( 𝐴 𝐹 𝑦 ∧ ∃! 𝑦 𝐴 𝐹 𝑦 ) → ( 𝐹𝐴 ) = 𝑦 )

Proof

Step Hyp Ref Expression
1 df-fv ( 𝐹𝐴 ) = ( ℩ 𝑦 𝐴 𝐹 𝑦 )
2 iota1 ( ∃! 𝑦 𝐴 𝐹 𝑦 → ( 𝐴 𝐹 𝑦 ↔ ( ℩ 𝑦 𝐴 𝐹 𝑦 ) = 𝑦 ) )
3 2 biimpac ( ( 𝐴 𝐹 𝑦 ∧ ∃! 𝑦 𝐴 𝐹 𝑦 ) → ( ℩ 𝑦 𝐴 𝐹 𝑦 ) = 𝑦 )
4 1 3 eqtrid ( ( 𝐴 𝐹 𝑦 ∧ ∃! 𝑦 𝐴 𝐹 𝑦 ) → ( 𝐹𝐴 ) = 𝑦 )