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Theorem ustexsym 20450
Description: In an uniform structure, for any entourage  V, there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
Assertion
Ref Expression
ustexsym  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  V ) )
Distinct variable groups:    w, U    w, V
Allowed substitution hint:    X( w)

Proof of Theorem ustexsym
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simplll 757 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  U  e.  (UnifOn `  X
) )
2 simplr 754 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  x  e.  U )
3 ustinvel 20444 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  `' x  e.  U )
41, 2, 3syl2anc 661 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  `' x  e.  U
)
5 ustincl 20442 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  `' x  e.  U  /\  x  e.  U )  ->  ( `' x  i^i  x )  e.  U
)
61, 4, 2, 5syl3anc 1228 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  -> 
( `' x  i^i  x )  e.  U
)
7 ustrel 20446 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  Rel  x )
8 dfrel2 5455 . . . . . . 7  |-  ( Rel  x  <->  `' `' x  =  x
)
97, 8sylib 196 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  `' `' x  =  x
)
109ineq1d 3699 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  ( `' `' x  i^i  `' x
)  =  ( x  i^i  `' x ) )
11 cnvin 5411 . . . . 5  |-  `' ( `' x  i^i  x
)  =  ( `' `' x  i^i  `' x
)
12 incom 3691 . . . . 5  |-  ( `' x  i^i  x )  =  ( x  i^i  `' x )
1310, 11, 123eqtr4g 2533 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  `' ( `' x  i^i  x
)  =  ( `' x  i^i  x ) )
141, 2, 13syl2anc 661 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  `' ( `' x  i^i  x )  =  ( `' x  i^i  x
) )
15 inss2 3719 . . . 4  |-  ( `' x  i^i  x ) 
C_  x
16 ustssco 20449 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  x  C_  ( x  o.  x
) )
171, 2, 16syl2anc 661 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  x  C_  ( x  o.  x ) )
18 simpr 461 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  -> 
( x  o.  x
)  C_  V )
1917, 18sstrd 3514 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  x  C_  V )
2015, 19syl5ss 3515 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  -> 
( `' x  i^i  x )  C_  V
)
21 cnveq 5174 . . . . . 6  |-  ( w  =  ( `' x  i^i  x )  ->  `' w  =  `' ( `' x  i^i  x
) )
22 id 22 . . . . . 6  |-  ( w  =  ( `' x  i^i  x )  ->  w  =  ( `' x  i^i  x ) )
2321, 22eqeq12d 2489 . . . . 5  |-  ( w  =  ( `' x  i^i  x )  ->  ( `' w  =  w  <->  `' ( `' x  i^i  x )  =  ( `' x  i^i  x
) ) )
24 sseq1 3525 . . . . 5  |-  ( w  =  ( `' x  i^i  x )  ->  (
w  C_  V  <->  ( `' x  i^i  x )  C_  V ) )
2523, 24anbi12d 710 . . . 4  |-  ( w  =  ( `' x  i^i  x )  ->  (
( `' w  =  w  /\  w  C_  V )  <->  ( `' ( `' x  i^i  x
)  =  ( `' x  i^i  x )  /\  ( `' x  i^i  x )  C_  V
) ) )
2625rspcev 3214 . . 3  |-  ( ( ( `' x  i^i  x )  e.  U  /\  ( `' ( `' x  i^i  x )  =  ( `' x  i^i  x )  /\  ( `' x  i^i  x
)  C_  V )
)  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  V ) )
276, 14, 20, 26syl12anc 1226 . 2  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  V ) )
28 ustexhalf 20445 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. x  e.  U  ( x  o.  x )  C_  V
)
2927, 28r19.29a 3003 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815    i^i cin 3475    C_ wss 3476   `'ccnv 4998    o. ccom 5003   Rel wrel 5004   ` cfv 5586  UnifOncust 20434
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 6574
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-rn 5010  df-res 5011  df-iota 5549  df-fun 5588  df-fv 5594  df-ust 20435
This theorem is referenced by:  ustex2sym  20451  neipcfilu  20531
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