Theorem tz6.12-1-afv

Description: Function value (Theorem 6.12(1) of TakeutiZaring p. 27, analogous to tz6.12-1 . (Contributed by Alexander van der Vekens, 29-Nov-2017)

	Ref	Expression
Assertion	tz6.12-1-afv	⊢ ( ( 𝐴 𝐹 𝑦 ∧ ∃! 𝑦 𝐴 𝐹 𝑦 ) → ( 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-afv	⊢ ( ( ⟨ 𝐴 , 𝑦 ⟩ ∈ 𝐹 ∧ ∃! 𝑦 ⟨ 𝐴 , 𝑦 ⟩ ∈ 𝐹 ) → ( 𝐹 ''' 𝐴 ) = 𝑦 )
4	1 2 3	syl2anb	⊢ ( ( 𝐴 𝐹 𝑦 ∧ ∃! 𝑦 𝐴 𝐹 𝑦 ) → ( 𝐹 ''' 𝐴 ) = 𝑦 )