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Theorem ustref 22742
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 2826 . . . . 5 𝐴 = 𝐴
2 resieq 5863 . . . . 5 ((𝐴𝑋𝐴𝑋) → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
31, 2mpbiri 259 . . . 4 ((𝐴𝑋𝐴𝑋) → 𝐴( I ↾ 𝑋)𝐴)
43anidms 567 . . 3 (𝐴𝑋𝐴( I ↾ 𝑋)𝐴)
543ad2ant3 1129 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴( I ↾ 𝑋)𝐴)
6 ustdiag 22732 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
76ssbrd 5106 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝐴( I ↾ 𝑋)𝐴𝐴𝑉𝐴))
873adant3 1126 . 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 396  w3a 1081   = wceq 1530  wcel 2107   class class class wbr 5063   I cid 5458  cres 5556  cfv 6352  UnifOncust 22723
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-res 5566  df-iota 6312  df-fun 6354  df-fv 6360  df-ust 22724
This theorem is referenced by:  cstucnd  22808
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