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Metamath Proof Explorer

Theorem glbcl

Description: The least upper bound function value belongs to the base set. (Contributed by NM, 7-Sep-2018)

Ref Expression
Hypotheses glbc.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
glbc.g ⊢ 𝐺 = ( glb ‘ 𝐾 )
glbc.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
glbc.s ⊢ ( 𝜑 → 𝑆 ∈ dom 𝐺 )
Assertion glbcl ( 𝜑 → ( 𝐺 ‘ 𝑆 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 glbc.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 glbc.g ⊢ 𝐺 = ( glb ‘ 𝐾 )
3 glbc.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
4 glbc.s ⊢ ( 𝜑 → 𝑆 ∈ dom 𝐺 )
5 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
6 biid ⊢ ( ( ∀ 𝑦 ∈ 𝑆 𝑥 ( le ‘ 𝐾 ) 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑧 ( le ‘ 𝐾 ) 𝑦 → 𝑧 ( le ‘ 𝐾 ) 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑆 𝑥 ( le ‘ 𝐾 ) 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑧 ( le ‘ 𝐾 ) 𝑦 → 𝑧 ( le ‘ 𝐾 ) 𝑥 ) ) )
7 1 5 2 3 4 glbelss ⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
8 1 5 2 6 3 7 glbval ⊢ ( 𝜑 → ( 𝐺 ‘ 𝑆 ) = ( ℩ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑥 ( le ‘ 𝐾 ) 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑧 ( le ‘ 𝐾 ) 𝑦 → 𝑧 ( le ‘ 𝐾 ) 𝑥 ) ) ) )
9 1 5 2 6 3 4 glbeu ⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑥 ( le ‘ 𝐾 ) 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑧 ( le ‘ 𝐾 ) 𝑦 → 𝑧 ( le ‘ 𝐾 ) 𝑥 ) ) )
10 riotacl ⊢ ( ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑥 ( le ‘ 𝐾 ) 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑧 ( le ‘ 𝐾 ) 𝑦 → 𝑧 ( le ‘ 𝐾 ) 𝑥 ) ) → ( ℩ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑥 ( le ‘ 𝐾 ) 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑧 ( le ‘ 𝐾 ) 𝑦 → 𝑧 ( le ‘ 𝐾 ) 𝑥 ) ) ) ∈ 𝐵 )
11 9 10 syl ⊢ ( 𝜑 → ( ℩ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑥 ( le ‘ 𝐾 ) 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑧 ( le ‘ 𝐾 ) 𝑦 → 𝑧 ( le ‘ 𝐾 ) 𝑥 ) ) ) ∈ 𝐵 )
12 8 11 eqeltrd ⊢ ( 𝜑 → ( 𝐺 ‘ 𝑆 ) ∈ 𝐵 )