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Theorem ustref 24136
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 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ 𝐴𝑉𝐴)
Proof of Theorem ustref
StepHypRef Expression
1 eqid 2728 . . . . 5 𝐴 = 𝐴
2 resieq 5996 . . . . 5 ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ (𝐴( I β†Ύ 𝑋)𝐴 ↔ 𝐴 = 𝐴))
31, 2mpbiri 258 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ 𝐴( I β†Ύ 𝑋)𝐴)
43anidms 566 . . 3 (𝐴 ∈ 𝑋 β†’ 𝐴( I β†Ύ 𝑋)𝐴)
543ad2ant3 1133 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ 𝐴( I β†Ύ 𝑋)𝐴)
6 ustdiag 24126 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ( I β†Ύ 𝑋) βŠ† 𝑉)
76ssbrd 5191 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝐴( I β†Ύ 𝑋)𝐴 β†’ 𝐴𝑉𝐴))
873adant3 1130 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ (𝐴( I β†Ύ 𝑋)𝐴 β†’ 𝐴𝑉𝐴))
95, 8mpd 15 1 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ 𝐴𝑉𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5148   I cid 5575   β†Ύ cres 5680  β€˜cfv 6548  UnifOncust 24117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-res 5690  df-iota 6500  df-fun 6550  df-fv 6556  df-ust 24118
This theorem is referenced by:  cstucnd  24202
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