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Theorem ustref 23723
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 2733 . . . . 5 𝐴 = 𝐴
2 resieq 5993 . . . . 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 568 . . 3 (𝐴 ∈ 𝑋 β†’ 𝐴( I β†Ύ 𝑋)𝐴)
543ad2ant3 1136 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ 𝐴( I β†Ύ 𝑋)𝐴)
6 ustdiag 23713 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ( I β†Ύ 𝑋) βŠ† 𝑉)
76ssbrd 5192 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝐴( I β†Ύ 𝑋)𝐴 β†’ 𝐴𝑉𝐴))
873adant3 1133 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5149   I cid 5574   β†Ύ cres 5679  β€˜cfv 6544  UnifOncust 23704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-ust 23705
This theorem is referenced by:  cstucnd  23789
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