Theorem suprubrnmpt2

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	suprubrnmpt2.x	⊢ Ⅎ 𝑥 𝜑
	suprubrnmpt2.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
	suprubrnmpt2.l	⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 )
	suprubrnmpt2.c	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
	suprubrnmpt2.d	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
	suprubrnmpt2.i	⊢ ( 𝑥 = 𝐶 → 𝐵 = 𝐷 )
Assertion	suprubrnmpt2	⊢ ( 𝜑 → 𝐷 ≤ sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ , < ) )

Proof

Step	Hyp	Ref	Expression
1		suprubrnmpt2.x	⊢ Ⅎ 𝑥 𝜑
2		suprubrnmpt2.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
3		suprubrnmpt2.l	⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 )
4		suprubrnmpt2.c	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
5		suprubrnmpt2.d	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
6		suprubrnmpt2.i	⊢ ( 𝑥 = 𝐶 → 𝐵 = 𝐷 )
7		eqid	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
8	1 7 2	rnmptssd	⊢ ( 𝜑 → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ )
9	7 6	elrnmpt1s	⊢ ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ ℝ ) → 𝐷 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
10	4 5 9	syl2anc	⊢ ( 𝜑 → 𝐷 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
11	10	ne0d	⊢ ( 𝜑 → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ≠ ∅ )
12	1 3	rnmptbdd	⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑤 ≤ 𝑦 )
13	8 11 12 10	suprubd	⊢ ( 𝜑 → 𝐷 ≤ sup ( ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) , ℝ , < ) )