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Theorem ustexsym 21010
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 760 . . . 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 21004 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  `' x  e.  U )
41, 2, 3syl2anc 659 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  `' x  e.  U
)
5 ustincl 21002 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  `' x  e.  U  /\  x  e.  U )  ->  ( `' x  i^i  x )  e.  U
)
61, 4, 2, 5syl3anc 1230 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  -> 
( `' x  i^i  x )  e.  U
)
7 ustrel 21006 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  Rel  x )
8 dfrel2 5274 . . . . . . 7  |-  ( Rel  x  <->  `' `' x  =  x
)
97, 8sylib 196 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  `' `' x  =  x
)
109ineq1d 3640 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  ( `' `' x  i^i  `' x
)  =  ( x  i^i  `' x ) )
11 cnvin 5231 . . . . 5  |-  `' ( `' x  i^i  x
)  =  ( `' `' x  i^i  `' x
)
12 incom 3632 . . . . 5  |-  ( `' x  i^i  x )  =  ( x  i^i  `' x )
1310, 11, 123eqtr4g 2468 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  `' ( `' x  i^i  x
)  =  ( `' x  i^i  x ) )
141, 2, 13syl2anc 659 . . 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 3660 . . . 4  |-  ( `' x  i^i  x ) 
C_  x
16 ustssco 21009 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  x  C_  ( x  o.  x
) )
171, 2, 16syl2anc 659 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  x  C_  ( x  o.  x ) )
18 simpr 459 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  -> 
( x  o.  x
)  C_  V )
1917, 18sstrd 3452 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  x  C_  V )
2015, 19syl5ss 3453 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  -> 
( `' x  i^i  x )  C_  V
)
21 cnveq 4997 . . . . . 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 2424 . . . . 5  |-  ( w  =  ( `' x  i^i  x )  ->  ( `' w  =  w  <->  `' ( `' x  i^i  x )  =  ( `' x  i^i  x
) ) )
24 sseq1 3463 . . . . 5  |-  ( w  =  ( `' x  i^i  x )  ->  (
w  C_  V  <->  ( `' x  i^i  x )  C_  V ) )
2523, 24anbi12d 709 . . . 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 3160 . . 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 1228 . 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 21005 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. x  e.  U  ( x  o.  x )  C_  V
)
2927, 28r19.29a 2949 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 367    = wceq 1405    e. wcel 1842   E.wrex 2755    i^i cin 3413    C_ wss 3414   `'ccnv 4822    o. ccom 4827   Rel wrel 4828   ` cfv 5569  UnifOncust 20994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-iota 5533  df-fun 5571  df-fv 5577  df-ust 20995
This theorem is referenced by:  ustex2sym  21011  neipcfilu  21091
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