Theorem supxrre3rnmpt

Description: The indexed supremum of a nonempty set of reals, is real if and only if it is bounded-above . (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	supxrre3rnmpt.x	⊢ Ⅎ 𝑥 𝜑
	supxrre3rnmpt.a	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
	supxrre3rnmpt.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
Assertion	supxrre3rnmpt	⊢ ( 𝜑 → ( sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ* , < ) ∈ ℝ ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		supxrre3rnmpt.x	⊢ Ⅎ 𝑥 𝜑
2		supxrre3rnmpt.a	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
3		supxrre3rnmpt.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
4		eqid	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
5	1 4 3	rnmptssd	⊢ ( 𝜑 → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ )
6	1 3 4 2	rnmptn0	⊢ ( 𝜑 → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ≠ ∅ )
7		supxrre3	⊢ ( ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ∧ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ≠ ∅ ) → ( sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ* , < ) ∈ ℝ ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑧 ≤ 𝑦 ) )
8	5 6 7	syl2anc	⊢ ( 𝜑 → ( sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ* , < ) ∈ ℝ ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑧 ≤ 𝑦 ) )
9	1 3	rnmptbd	⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑧 ≤ 𝑦 ) )
10	8 9	bitr4d	⊢ ( 𝜑 → ( sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ* , < ) ∈ ℝ ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) )