Theorem ustref

Description: Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017)

	Ref	Expression
Assertion	ustref	⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋 ) → 𝐴 𝑉 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 𝐴 = 𝐴
2		resieq	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 ( I ↾ 𝑋 ) 𝐴 ↔ 𝐴 = 𝐴 ) )
3	1 2	mpbiri	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ( I ↾ 𝑋 ) 𝐴 )
4	3	anidms	⊢ ( 𝐴 ∈ 𝑋 → 𝐴 ( I ↾ 𝑋 ) 𝐴 )
5	4	3ad2ant3	⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ( I ↾ 𝑋 ) 𝐴 )
6		ustdiag	⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ 𝑈 ) → ( I ↾ 𝑋 ) ⊆ 𝑉 )
7	6	ssbrd	⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ 𝑈 ) → ( 𝐴 ( I ↾ 𝑋 ) 𝐴 → 𝐴 𝑉 𝐴 ) )
8	7	3adant3	⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 ( I ↾ 𝑋 ) 𝐴 → 𝐴 𝑉 𝐴 ) )
9	5 8	mpd	⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋 ) → 𝐴 𝑉 𝐴 )