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Theorem ustrel 20477
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 20470 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X
) )
2 xpss 5109 . . 3  |-  ( X  X.  X )  C_  ( _V  X.  _V )
31, 2syl6ss 3516 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( _V  X.  _V ) )
4 df-rel 5006 . 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 1767   _Vcvv 3113    C_ wss 3476    X. cxp 4997   Rel wrel 5004   ` cfv 5588  UnifOncust 20465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fv 5596  df-ust 20466
This theorem is referenced by:  ustssco  20480  ustexsym  20481  ustuqtop4  20510  utop2nei  20516  utop3cls  20517  ucncn  20551
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