Theorem pimltpnf

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, 20-Dec-2024)

	Ref	Expression
Hypotheses	pimltpnf.1	$$⊢Ⅎx\phi$$
	pimltpnf.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℝ$$
Assertion	pimltpnf	$$⊢\phi \to \left\{x\in A|B<\mathrm{+\infty }\right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		pimltpnf.1	$$⊢Ⅎx\phi$$
2		pimltpnf.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℝ$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
4	1 3 2	pimltpnff	$$⊢\phi \to \left\{x\in A|B<\mathrm{+\infty }\right\}=A$$