Theorem glbelss

Description: A member of the domain of the greatest lower bound function is a subset of the base set. (Contributed by NM, 7-Sep-2018)

	Ref	Expression
Hypotheses	glbs.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	glbs.l	⊢ ≤ = ( le ‘ 𝐾 )
	glbs.g	⊢ 𝐺 = ( glb ‘ 𝐾 )
	glbs.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
	glbs.s	⊢ ( 𝜑 → 𝑆 ∈ dom 𝐺 )
Assertion	glbelss	⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		glbs.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		glbs.l	⊢ ≤ = ( le ‘ 𝐾 )
3		glbs.g	⊢ 𝐺 = ( glb ‘ 𝐾 )
4		glbs.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
5		glbs.s	⊢ ( 𝜑 → 𝑆 ∈ dom 𝐺 )
6		biid	⊢ ( ( ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥 ) ) )
7	1 2 3 6 4	glbeldm	⊢ ( 𝜑 → ( 𝑆 ∈ dom 𝐺 ↔ ( 𝑆 ⊆ 𝐵 ∧ ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥 ) ) ) ) )
8	5 7	mpbid	⊢ ( 𝜑 → ( 𝑆 ⊆ 𝐵 ∧ ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥 ) ) ) )
9	8	simpld	⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )