Theorem pimltpnf2

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

	Ref	Expression
Hypotheses	pimltpnf2.1	|- F/_ x F
	pimltpnf2.2	|- ( ph -> F : A --> RR )
Assertion	pimltpnf2	|- ( ph -> { x e. A | ( F ` x ) < +oo } = A )

Proof

Step	Hyp	Ref	Expression
1		pimltpnf2.1	|- F/_ x F
2		pimltpnf2.2	|- ( ph -> F : A --> RR )
3		nfcv	|- F/_ x A
4	1 3 2	pimltpnf2f	|- ( ph -> { x e. A | ( F ` x ) < +oo } = A )