Theorem tz6.12c

Description: Corollary of Theorem 6.12(1) of TakeutiZaring p. 27. (Contributed by NM, 30-Apr-2004) (Proof shortened by SN, 23-Dec-2024)

	Ref	Expression
Assertion	tz6.12c	⊢ ( ∃! 𝑦 𝐴 𝐹 𝑦 → ( ( 𝐹 ‘ 𝐴 ) = 𝑦 ↔ 𝐴 𝐹 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		df-fv	⊢ ( 𝐹 ‘ 𝐴 ) = ( ℩ 𝑦 𝐴 𝐹 𝑦 )
2	1	eqeq1i	⊢ ( ( 𝐹 ‘ 𝐴 ) = 𝑦 ↔ ( ℩ 𝑦 𝐴 𝐹 𝑦 ) = 𝑦 )
3		iota1	⊢ ( ∃! 𝑦 𝐴 𝐹 𝑦 → ( 𝐴 𝐹 𝑦 ↔ ( ℩ 𝑦 𝐴 𝐹 𝑦 ) = 𝑦 ) )
4	2 3	bitr4id	⊢ ( ∃! 𝑦 𝐴 𝐹 𝑦 → ( ( 𝐹 ‘ 𝐴 ) = 𝑦 ↔ 𝐴 𝐹 𝑦 ) )