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Theorem ustssco 21221
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 3630 . . . 4  |-  V  C_  ( V  u.  ( V  o.  V )
)
2 coires1 5370 . . . . . 6  |-  ( V  o.  (  _I  |`  X ) )  =  ( V  |`  X )
3 ustrel 21218 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )
4 ustssxp 21211 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X
) )
5 dmss 5051 . . . . . . . . 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 5071 . . . . . . . 8  |-  dom  ( X  X.  X )  =  X
86, 7syl6sseq 3511 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  dom  V 
C_  X )
9 relssres 5159 . . . . . . 7  |-  ( ( Rel  V  /\  dom  V 
C_  X )  -> 
( V  |`  X )  =  V )
103, 8, 9syl2anc 666 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  ( V  |`  X )  =  V )
112, 10syl5eq 2476 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  ( V  o.  (  _I  |`  X ) )  =  V )
1211uneq1d 3620 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (
( V  o.  (  _I  |`  X ) )  u.  ( V  o.  V ) )  =  ( V  u.  ( V  o.  V )
) )
131, 12syl5sseqr 3514 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( ( V  o.  (  _I  |`  X ) )  u.  ( V  o.  V ) ) )
14 coundi 5353 . . 3  |-  ( V  o.  ( (  _I  |`  X )  u.  V
) )  =  ( ( V  o.  (  _I  |`  X ) )  u.  ( V  o.  V ) )
1513, 14syl6sseqr 3512 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( V  o.  (
(  _I  |`  X )  u.  V ) ) )
16 ustdiag 21215 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (  _I  |`  X )  C_  V )
17 ssequn1 3637 . . . 4  |-  ( (  _I  |`  X )  C_  V  <->  ( (  _I  |`  X )  u.  V
)  =  V )
1816, 17sylib 200 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (
(  _I  |`  X )  u.  V )  =  V )
1918coeq2d 5014 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  ( V  o.  ( (  _I  |`  X )  u.  V ) )  =  ( V  o.  V
) )
2015, 19sseqtrd 3501 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( V  o.  V
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1869    u. cun 3435    C_ wss 3437    _I cid 4761    X. cxp 4849   dom cdm 4851    |` cres 4853    o. ccom 4855   Rel wrel 4856   ` cfv 5599  UnifOncust 21206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-iota 5563  df-fun 5601  df-fv 5607  df-ust 21207
This theorem is referenced by:  ustexsym  21222  ustex3sym  21224
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