Theorem elbl

Description: Membership in a ball. (Contributed by NM, 2-Sep-2006) (Revised by Mario Carneiro, 11-Nov-2013)

	Ref	Expression
Assertion	elbl	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		blval	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) = { 𝑥 ∈ 𝑋 ∣ ( 𝑃 𝐷 𝑥 ) < 𝑅 } )
2	1	eleq2d	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ 𝐴 ∈ { 𝑥 ∈ 𝑋 ∣ ( 𝑃 𝐷 𝑥 ) < 𝑅 } ) )
3		oveq2	⊢ ( 𝑥 = 𝐴 → ( 𝑃 𝐷 𝑥 ) = ( 𝑃 𝐷 𝐴 ) )
4	3	breq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑃 𝐷 𝑥 ) < 𝑅 ↔ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) )
5	4	elrab	⊢ ( 𝐴 ∈ { 𝑥 ∈ 𝑋 ∣ ( 𝑃 𝐷 𝑥 ) < 𝑅 } ↔ ( 𝐴 ∈ 𝑋 ∧ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) )
6	2 5	syl6bb	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) ) )