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 )