Theorem luble

Description: The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011) (Revised by NM, 7-Sep-2018)

	Ref	Expression
Hypotheses	lubprop.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	lubprop.l	⊢ ≤ = ( le ‘ 𝐾 )
	lubprop.u	⊢ 𝑈 = ( lub ‘ 𝐾 )
	lubprop.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
	lubprop.s	⊢ ( 𝜑 → 𝑆 ∈ dom 𝑈 )
	luble.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
Assertion	luble	⊢ ( 𝜑 → 𝑋 ≤ ( 𝑈 ‘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		lubprop.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		lubprop.l	⊢ ≤ = ( le ‘ 𝐾 )
3		lubprop.u	⊢ 𝑈 = ( lub ‘ 𝐾 )
4		lubprop.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
5		lubprop.s	⊢ ( 𝜑 → 𝑆 ∈ dom 𝑈 )
6		luble.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
7		breq1	⊢ ( 𝑦 = 𝑋 → ( 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ↔ 𝑋 ≤ ( 𝑈 ‘ 𝑆 ) ) )
8	1 2 3 4 5	lubprop	⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) )
9	8	simpld	⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) )
10	7 9 6	rspcdva	⊢ ( 𝜑 → 𝑋 ≤ ( 𝑈 ‘ 𝑆 ) )