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	$$⊢\underset{\_}{Ⅎ}xF$$
	pimltpnf2.2	$$⊢\phi \to F:A⟶ℝ$$
Assertion	pimltpnf2	$$⊢\phi \to \left\{x\in A|F\left(x\right)<\mathrm{+\infty }\right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		pimltpnf2.1	$$⊢\underset{\_}{Ⅎ}xF$$
2		pimltpnf2.2	$$⊢\phi \to F:A⟶ℝ$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
4	1 3 2	pimltpnf2f	$$⊢\phi \to \left\{x\in A|F\left(x\right)<\mathrm{+\infty }\right\}=A$$