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Metamath Proof Explorer

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 )