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Theorem ustrel 19784
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 19777 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X
) )
2 xpss 4944 . . 3  |-  ( X  X.  X )  C_  ( _V  X.  _V )
31, 2syl6ss 3366 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( _V  X.  _V ) )
4 df-rel 4845 . 2  |-  ( Rel 
V  <->  V  C_  ( _V 
X.  _V ) )
53, 4sylibr 212 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   _Vcvv 2970    C_ wss 3326    X. cxp 4836   Rel wrel 4843   ` cfv 5416  UnifOncust 19772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-res 4850  df-iota 5379  df-fun 5418  df-fv 5424  df-ust 19773
This theorem is referenced by:  ustssco  19787  ustexsym  19788  ustuqtop4  19817  utop2nei  19823  utop3cls  19824  ucncn  19858
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