Theorem pimltmnf2

Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -oo , is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021) (Revised by Glauco Siliprandi, 15-Dec-2024)

	Ref	Expression
Hypotheses	pimltmnf2.1	|- F/_ x F
	pimltmnf2.2	|- ( ph -> F : A --> RR )
Assertion	pimltmnf2	|- ( ph -> { x e. A | ( F ` x ) < -oo } = (/) )

Proof

Step	Hyp	Ref	Expression
1		pimltmnf2.1	|- F/_ x F
2		pimltmnf2.2	|- ( ph -> F : A --> RR )
3		nfcv	|- F/_ x A
4	1 3 2	pimltmnf2f	|- ( ph -> { x e. A | ( F ` x ) < -oo } = (/) )