MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustref Structured version   Visualization version   GIF version
Theorem ustref 24067
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 2724 . . . . 5 𝐴 = 𝐴
2 resieq 5983 . . . . 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 1132 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ 𝐴( I β†Ύ 𝑋)𝐴)
6 ustdiag 24057 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ( I β†Ύ 𝑋) βŠ† 𝑉)
76ssbrd 5182 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝐴( I β†Ύ 𝑋)𝐴 β†’ 𝐴𝑉𝐴))
873adant3 1129 . 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 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5139   I cid 5564   β†Ύ cres 5669  β€˜cfv 6534  UnifOncust 24048
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 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-res 5679  df-iota 6486  df-fun 6536  df-fv 6542  df-ust 24049
This theorem is referenced by:  cstucnd  24133
  Copyright terms: Public domain W3C validator