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Metamath Proof Explorer

Theorem elrn2g

Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011)

		Ref	Expression
	Assertion	elrn2g	$⊢ A \in V \to (A \in ran B \leftrightarrow \exists x ⟨x, A⟩ \in B)$

Step	Hyp	Ref	Expression
1		opeq2	$⊢ y = A \to ⟨x, y⟩ = ⟨x, A⟩$
2	1	eleq1d	$⊢ y = A \to (⟨x, y⟩ \in B \leftrightarrow ⟨x, A⟩ \in B)$
3	2	exbidv	$⊢ y = A \to (\exists x ⟨x, y⟩ \in B \leftrightarrow \exists x ⟨x, A⟩ \in B)$
4		dfrn3	$⊢ ran B = \{y \| \exists x ⟨x, y⟩ \in B\}$
5	3 4	elab2g	$⊢ A \in V \to (A \in ran B \leftrightarrow \exists x ⟨x, A⟩ \in B)$