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Theorem ustref 24172
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 2725 . . . . 5 𝐴 = 𝐴
2 resieq 5996 . . . . 5 ((𝐴𝑋𝐴𝑋) → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
31, 2mpbiri 257 . . . 4 ((𝐴𝑋𝐴𝑋) → 𝐴( I ↾ 𝑋)𝐴)
43anidms 565 . . 3 (𝐴𝑋𝐴( I ↾ 𝑋)𝐴)
543ad2ant3 1132 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴( I ↾ 𝑋)𝐴)
6 ustdiag 24162 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
76ssbrd 5192 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝐴( I ↾ 𝑋)𝐴𝐴𝑉𝐴))
873adant3 1129 . 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 394  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5149   I cid 5575  cres 5680  cfv 6549  UnifOncust 24153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-res 5690  df-iota 6501  df-fun 6551  df-fv 6557  df-ust 24154
This theorem is referenced by:  cstucnd  24238
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