Theorem fimaxre4

Description: A nonempty finite set of real numbers is bounded (image set version). (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	fimaxre4.1	⊢ Ⅎ 𝑥 𝜑
	fimaxre4.2	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fimaxre4.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
Assertion	fimaxre4	⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		fimaxre4.1	⊢ Ⅎ 𝑥 𝜑
2		fimaxre4.2	⊢ ( 𝜑 → 𝐴 ∈ Fin )
3		fimaxre4.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
4	3	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝐵 ∈ ℝ ) )
5	1 4	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ )
6		fimaxre3	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 )
7	2 5 6	syl2anc	⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 )