Theorem suprubrnmpt

Description: A member of a nonempty indexed set of reals is less than or equal to the set's upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	suprubrnmpt.x	⊢ Ⅎ 𝑥 𝜑
	suprubrnmpt.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
	suprubrnmpt.e	⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 )
Assertion	suprubrnmpt	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ , < ) )

Proof

Step	Hyp	Ref	Expression
1		suprubrnmpt.x	⊢ Ⅎ 𝑥 𝜑
2		suprubrnmpt.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
3		suprubrnmpt.e	⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 )
4		eqid	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
5	1 4 2	rnmptssd	⊢ ( 𝜑 → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ )
6	5	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ )
7		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 )
8	4	elrnmpt1	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ ) → 𝐵 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
9	7 2 8	syl2anc	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
10	9	ne0d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ≠ ∅ )
11	1 3	rnmptbdd	⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑤 ≤ 𝑦 )
12	11	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑤 ≤ 𝑦 )
13	6 10 12 9	suprubd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ , < ) )