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Theorem ustref 22219
 Description: Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
ustref ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴𝑉𝐴)

Proof of Theorem ustref
StepHypRef Expression
1 eqid 2756 . . . . 5 𝐴 = 𝐴
2 resieq 5561 . . . . 5 ((𝐴𝑋𝐴𝑋) → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
31, 2mpbiri 248 . . . 4 ((𝐴𝑋𝐴𝑋) → 𝐴( I ↾ 𝑋)𝐴)
43anidms 680 . . 3 (𝐴𝑋𝐴( I ↾ 𝑋)𝐴)
543ad2ant3 1130 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴( I ↾ 𝑋)𝐴)
6 ustdiag 22209 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
76ssbrd 4843 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝐴( I ↾ 𝑋)𝐴𝐴𝑉𝐴))
873adant3 1127 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → (𝐴( I ↾ 𝑋)𝐴𝐴𝑉𝐴))
95, 8mpd 15 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴𝑉𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1628   ∈ wcel 2135   class class class wbr 4800   I cid 5169   ↾ cres 5264  ‘cfv 6045  UnifOncust 22200 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-res 5274  df-iota 6008  df-fun 6047  df-fv 6053  df-ust 22201 This theorem is referenced by:  cstucnd  22285
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