Metamath Proof Explorer

Theorem dmfexALT

Description: Alternate proof of dmfex : shorter but using ax-rep . (Contributed by NM, 27-Aug-2006) (Proof shortened by Andrew Salmon, 17-Sep-2011) (Proof shortened by AV, 23-Aug-2024) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	dmfexALT	$⊢ (F \in C \land F : A ⟶ B) \to A \in V$

Proof

Step	Hyp	Ref	Expression
1		elex	$⊢ F \in C \to F \in V$
2		fdmexb	$⊢ F : A ⟶ B \to (A \in V \leftrightarrow F \in V)$
3	2	biimprd	$⊢ F : A ⟶ B \to (F \in V \to A \in V)$
4	1 3	mpan9	$⊢ (F \in C \land F : A ⟶ B) \to A \in V$