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 ) → ( 𝐹 ‘ 𝐴 ) = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		df-fv	⊢ ( 𝐹 ‘ 𝐴 ) = ( ℩ 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 ) → ( 𝐹 ‘ 𝐴 ) = 𝑦 )