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	$$⊢\underset{\_}{Ⅎ}xF$$
	pimgtpnf2.2	$$⊢\phi \to F:A⟶ℝ$$
Assertion	pimgtpnf2	$$⊢\phi \to \left\{x\in A|\mathrm{+\infty }<F\left(x\right)\right\}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		pimgtpnf2.1	$$⊢\underset{\_}{Ⅎ}xF$$
2		pimgtpnf2.2	$$⊢\phi \to F:A⟶ℝ$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
4	1 3 2	pimgtpnf2f	$$⊢\phi \to \left\{x\in A|\mathrm{+\infty }<F\left(x\right)\right\}=\varnothing$$