MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ussid Structured version   Unicode version

Theorem ussid 19960
Description: In case the base of the UnifSt element of the uniform space is the base of its element structure, then UnifSt does not restrict it further. (Contributed by Thierry Arnoux, 4-Dec-2017.)
Hypotheses
Ref Expression
ussval.1  |-  B  =  ( Base `  W
)
ussval.2  |-  U  =  ( UnifSet `  W )
Assertion
Ref Expression
ussid  |-  ( ( B  X.  B )  =  U. U  ->  U  =  (UnifSt `  W
) )

Proof of Theorem ussid
StepHypRef Expression
1 oveq2 6201 . . 3  |-  ( ( B  X.  B )  =  U. U  -> 
( Ut  ( B  X.  B ) )  =  ( Ut  U. U ) )
2 id 22 . . . . . 6  |-  ( ( B  X.  B )  =  U. U  -> 
( B  X.  B
)  =  U. U
)
3 ussval.1 . . . . . . . 8  |-  B  =  ( Base `  W
)
4 fvex 5802 . . . . . . . 8  |-  ( Base `  W )  e.  _V
53, 4eqeltri 2535 . . . . . . 7  |-  B  e. 
_V
65, 5xpex 6611 . . . . . 6  |-  ( B  X.  B )  e. 
_V
72, 6syl6eqelr 2548 . . . . 5  |-  ( ( B  X.  B )  =  U. U  ->  U. U  e.  _V )
8 uniexb 6489 . . . . 5  |-  ( U  e.  _V  <->  U. U  e. 
_V )
97, 8sylibr 212 . . . 4  |-  ( ( B  X.  B )  =  U. U  ->  U  e.  _V )
10 eqid 2451 . . . . 5  |-  U. U  =  U. U
1110restid 14483 . . . 4  |-  ( U  e.  _V  ->  ( Ut  U. U )  =  U )
129, 11syl 16 . . 3  |-  ( ( B  X.  B )  =  U. U  -> 
( Ut  U. U )  =  U )
131, 12eqtr2d 2493 . 2  |-  ( ( B  X.  B )  =  U. U  ->  U  =  ( Ut  ( B  X.  B ) ) )
14 ussval.2 . . 3  |-  U  =  ( UnifSet `  W )
153, 14ussval 19959 . 2  |-  ( Ut  ( B  X.  B ) )  =  (UnifSt `  W )
1613, 15syl6eq 2508 1  |-  ( ( B  X.  B )  =  U. U  ->  U  =  (UnifSt `  W
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3071   U.cuni 4192    X. cxp 4939   ` cfv 5519  (class class class)co 6193   Basecbs 14285   UnifSetcunif 14359   ↾t crest 14470  UnifStcuss 19953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-rest 14472  df-uss 19956
This theorem is referenced by:  tususs  19970  cmetcusp  20991  cnflduss  20993
  Copyright terms: Public domain W3C validator