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Theorem ustrel 21157
Description: The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)
Assertion
Ref Expression
ustrel  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )

Proof of Theorem ustrel
StepHypRef Expression
1 ustssxp 21150 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X
) )
2 xpss 4961 . . 3  |-  ( X  X.  X )  C_  ( _V  X.  _V )
31, 2syl6ss 3482 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( _V  X.  _V ) )
4 df-rel 4861 . 2  |-  ( Rel 
V  <->  V  C_  ( _V 
X.  _V ) )
53, 4sylibr 215 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1870   _Vcvv 3087    C_ wss 3442    X. cxp 4852   Rel wrel 4859   ` cfv 5601  UnifOncust 21145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-ust 21146
This theorem is referenced by:  ustssco  21160  ustexsym  21161  ustuqtop4  21190  utop2nei  21196  utop3cls  21197  ucncn  21231
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