Theorem pimltmnf2

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

	Ref	Expression
Hypotheses	pimltmnf2.1	⊢ Ⅎ 𝑥 𝐹
	pimltmnf2.2	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ )
Assertion	pimltmnf2	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < -∞ } = ∅ )

Proof

Step	Hyp	Ref	Expression
1		pimltmnf2.1	⊢ Ⅎ 𝑥 𝐹
2		pimltmnf2.2	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ )
3		nfcv	⊢ Ⅎ 𝑥 𝐴
4	1 3 2	pimltmnf2f	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) < -∞ } = ∅ )