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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	$⊢ (⟨A, y⟩ \in F \land \exists! y ⟨A, y⟩ \in F) \to F (A) = y$

Step	Hyp	Ref	Expression
1		df-br	$⊢ A F y \leftrightarrow ⟨A, y⟩ \in F$
2	1	eubii	$⊢ \exists! y A F y \leftrightarrow \exists! y ⟨A, y⟩ \in F$
3		tz6.12-1	$⊢ (A F y \land \exists! y A F y) \to F (A) = y$
4	1 2 3	syl2anbr	$⊢ (⟨A, y⟩ \in F \land \exists! y ⟨A, y⟩ \in F) \to F (A) = y$