Metamath Proof Explorer


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 ( 𝜑𝐾𝑉 )
glbs.s ( 𝜑𝑆 ∈ dom 𝐺 )
Assertion glbelss ( 𝜑𝑆𝐵 )

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 ( 𝜑𝐾𝑉 )
5 glbs.s ( 𝜑𝑆 ∈ dom 𝐺 )
6 biid ( ( ∀ 𝑦𝑆 𝑥 𝑦 ∧ ∀ 𝑧𝐵 ( ∀ 𝑦𝑆 𝑧 𝑦𝑧 𝑥 ) ) ↔ ( ∀ 𝑦𝑆 𝑥 𝑦 ∧ ∀ 𝑧𝐵 ( ∀ 𝑦𝑆 𝑧 𝑦𝑧 𝑥 ) ) )
7 1 2 3 6 4 glbeldm ( 𝜑 → ( 𝑆 ∈ dom 𝐺 ↔ ( 𝑆𝐵 ∧ ∃! 𝑥𝐵 ( ∀ 𝑦𝑆 𝑥 𝑦 ∧ ∀ 𝑧𝐵 ( ∀ 𝑦𝑆 𝑧 𝑦𝑧 𝑥 ) ) ) ) )
8 5 7 mpbid ( 𝜑 → ( 𝑆𝐵 ∧ ∃! 𝑥𝐵 ( ∀ 𝑦𝑆 𝑥 𝑦 ∧ ∀ 𝑧𝐵 ( ∀ 𝑦𝑆 𝑧 𝑦𝑧 𝑥 ) ) ) )
9 8 simpld ( 𝜑𝑆𝐵 )