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Metamath Proof Explorer

Theorem rnfvprc

Description: The range of a function value at a proper class is empty. (Contributed by AV, 20-Aug-2022)

		Ref	Expression
	Hypothesis	rnfvprc.y	$⊢ Y = F (X)$
	Assertion	rnfvprc	$⊢ \neg X \in V \to ran Y = \emptyset$

Step	Hyp	Ref	Expression
1		rnfvprc.y	$⊢ Y = F (X)$
2		fvprc	$⊢ \neg X \in V \to F (X) = \emptyset$
3	1 2	eqtrid	$⊢ \neg X \in V \to Y = \emptyset$
4	3	rneqd	$⊢ \neg X \in V \to ran Y = ran \emptyset$
5		rn0	$⊢ ran \emptyset = \emptyset$
6	4 5	eqtrdi	$⊢ \neg X \in V \to ran Y = \emptyset$