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Theorem ustssco 20899
Description: In an uniform structure, any entourage  V is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)
Assertion
Ref Expression
ustssco  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( V  o.  V
) )

Proof of Theorem ustssco
StepHypRef Expression
1 ssun1 3603 . . . 4  |-  V  C_  ( V  u.  ( V  o.  V )
)
2 coires1 5460 . . . . . 6  |-  ( V  o.  (  _I  |`  X ) )  =  ( V  |`  X )
3 ustrel 20896 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )
4 ustssxp 20889 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X
) )
5 dmss 5142 . . . . . . . . 9  |-  ( V 
C_  ( X  X.  X )  ->  dom  V 
C_  dom  ( X  X.  X ) )
64, 5syl 17 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  dom  V 
C_  dom  ( X  X.  X ) )
7 dmxpid 5162 . . . . . . . 8  |-  dom  ( X  X.  X )  =  X
86, 7syl6sseq 3485 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  dom  V 
C_  X )
9 relssres 5250 . . . . . . 7  |-  ( ( Rel  V  /\  dom  V 
C_  X )  -> 
( V  |`  X )  =  V )
103, 8, 9syl2anc 659 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  ( V  |`  X )  =  V )
112, 10syl5eq 2453 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  ( V  o.  (  _I  |`  X ) )  =  V )
1211uneq1d 3593 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (
( V  o.  (  _I  |`  X ) )  u.  ( V  o.  V ) )  =  ( V  u.  ( V  o.  V )
) )
131, 12syl5sseqr 3488 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( ( V  o.  (  _I  |`  X ) )  u.  ( V  o.  V ) ) )
14 coundi 5443 . . 3  |-  ( V  o.  ( (  _I  |`  X )  u.  V
) )  =  ( ( V  o.  (  _I  |`  X ) )  u.  ( V  o.  V ) )
1513, 14syl6sseqr 3486 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( V  o.  (
(  _I  |`  X )  u.  V ) ) )
16 ustdiag 20893 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (  _I  |`  X )  C_  V )
17 ssequn1 3610 . . . 4  |-  ( (  _I  |`  X )  C_  V  <->  ( (  _I  |`  X )  u.  V
)  =  V )
1816, 17sylib 196 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (
(  _I  |`  X )  u.  V )  =  V )
1918coeq2d 5105 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  ( V  o.  ( (  _I  |`  X )  u.  V ) )  =  ( V  o.  V
) )
2015, 19sseqtrd 3475 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( V  o.  V
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840    u. cun 3409    C_ wss 3411    _I cid 4730    X. cxp 4938   dom cdm 4940    |` cres 4942    o. ccom 4944   Rel wrel 4945   ` cfv 5523  UnifOncust 20884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-iota 5487  df-fun 5525  df-fv 5531  df-ust 20885
This theorem is referenced by:  ustexsym  20900  ustex3sym  20902
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