Theorem totbndss

Description: A subset of a totally bounded metric space is totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009) (Revised by Mario Carneiro, 12-Sep-2015)

	Ref	Expression
Assertion	totbndss	⊢ ( ( 𝑀 ∈ ( TotBnd ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑀 ↾ ( 𝑆 × 𝑆 ) ) ∈ ( TotBnd ‘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		istotbnd	⊢ ( 𝑀 ∈ ( TotBnd ‘ 𝑋 ) ↔ ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ ∀ 𝑑 ∈ ℝ+ ∃ 𝑣 ∈ Fin ( ∪ 𝑣 = 𝑋 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) ) )
2	1	simprbi	⊢ ( 𝑀 ∈ ( TotBnd ‘ 𝑋 ) → ∀ 𝑑 ∈ ℝ+ ∃ 𝑣 ∈ Fin ( ∪ 𝑣 = 𝑋 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) )
3		sseq2	⊢ ( ∪ 𝑣 = 𝑋 → ( 𝑆 ⊆ ∪ 𝑣 ↔ 𝑆 ⊆ 𝑋 ) )
4	3	biimprcd	⊢ ( 𝑆 ⊆ 𝑋 → ( ∪ 𝑣 = 𝑋 → 𝑆 ⊆ ∪ 𝑣 ) )
5	4	anim1d	⊢ ( 𝑆 ⊆ 𝑋 → ( ( ∪ 𝑣 = 𝑋 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) → ( 𝑆 ⊆ ∪ 𝑣 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) ) )
6	5	reximdv	⊢ ( 𝑆 ⊆ 𝑋 → ( ∃ 𝑣 ∈ Fin ( ∪ 𝑣 = 𝑋 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) → ∃ 𝑣 ∈ Fin ( 𝑆 ⊆ ∪ 𝑣 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) ) )
7	6	ralimdv	⊢ ( 𝑆 ⊆ 𝑋 → ( ∀ 𝑑 ∈ ℝ+ ∃ 𝑣 ∈ Fin ( ∪ 𝑣 = 𝑋 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) → ∀ 𝑑 ∈ ℝ+ ∃ 𝑣 ∈ Fin ( 𝑆 ⊆ ∪ 𝑣 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) ) )
8	2 7	mpan9	⊢ ( ( 𝑀 ∈ ( TotBnd ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ∀ 𝑑 ∈ ℝ+ ∃ 𝑣 ∈ Fin ( 𝑆 ⊆ ∪ 𝑣 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) )
9		totbndmet	⊢ ( 𝑀 ∈ ( TotBnd ‘ 𝑋 ) → 𝑀 ∈ ( Met ‘ 𝑋 ) )
10		eqid	⊢ ( 𝑀 ↾ ( 𝑆 × 𝑆 ) ) = ( 𝑀 ↾ ( 𝑆 × 𝑆 ) )
11	10	sstotbnd	⊢ ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( ( 𝑀 ↾ ( 𝑆 × 𝑆 ) ) ∈ ( TotBnd ‘ 𝑆 ) ↔ ∀ 𝑑 ∈ ℝ+ ∃ 𝑣 ∈ Fin ( 𝑆 ⊆ ∪ 𝑣 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) ) )
12	9 11	sylan	⊢ ( ( 𝑀 ∈ ( TotBnd ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( ( 𝑀 ↾ ( 𝑆 × 𝑆 ) ) ∈ ( TotBnd ‘ 𝑆 ) ↔ ∀ 𝑑 ∈ ℝ+ ∃ 𝑣 ∈ Fin ( 𝑆 ⊆ ∪ 𝑣 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) ) )
13	8 12	mpbird	⊢ ( ( 𝑀 ∈ ( TotBnd ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑀 ↾ ( 𝑆 × 𝑆 ) ) ∈ ( TotBnd ‘ 𝑆 ) )