Theorem pimgtmnf

Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -oo , is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021) (Revised by Glauco Siliprandi, 20-Dec-2024)

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

Proof

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