Theorem pimgtpnf2

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

	Ref	Expression
Hypotheses	pimgtpnf2.1	|- F/_ x F
	pimgtpnf2.2	|- ( ph -> F : A --> RR )
Assertion	pimgtpnf2	|- ( ph -> { x e. A | +oo < ( F ` x ) } = (/) )

Proof

Step	Hyp	Ref	Expression
1		pimgtpnf2.1	|- F/_ x F
2		pimgtpnf2.2	|- ( ph -> F : A --> RR )
3		nfcv	|- F/_ x A
4	1 3 2	pimgtpnf2f	|- ( ph -> { x e. A | +oo < ( F ` x ) } = (/) )