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	⊢ Ⅎ 𝑥 𝐹
	pimgtpnf2.2	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ )
Assertion	pimgtpnf2	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ +∞ < ( 𝐹 ‘ 𝑥 ) } = ∅ )

Proof

Step	Hyp	Ref	Expression
1		pimgtpnf2.1	⊢ Ⅎ 𝑥 𝐹
2		pimgtpnf2.2	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ )
3		nfcv	⊢ Ⅎ 𝑥 𝐴
4	1 3 2	pimgtpnf2f	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ +∞ < ( 𝐹 ‘ 𝑥 ) } = ∅ )