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

Theorem ussval 21266
Description: The uniform structure on uniform space  W. This proof uses a trick with fvprc 5873 to avoid requiring  W to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.)
Hypotheses
Ref Expression
ussval.1  |-  B  =  ( Base `  W
)
ussval.2  |-  U  =  ( UnifSet `  W )
Assertion
Ref Expression
ussval  |-  ( Ut  ( B  X.  B ) )  =  (UnifSt `  W )

Proof of Theorem ussval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 fveq2 5879 . . . . 5  |-  ( w  =  W  ->  ( UnifSet
`  w )  =  ( UnifSet `  W )
)
2 fveq2 5879 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
32sqxpeqd 4877 . . . . 5  |-  ( w  =  W  ->  (
( Base `  w )  X.  ( Base `  w
) )  =  ( ( Base `  W
)  X.  ( Base `  W ) ) )
41, 3oveq12d 6321 . . . 4  |-  ( w  =  W  ->  (
( UnifSet `  w )t  (
( Base `  w )  X.  ( Base `  w
) ) )  =  ( ( UnifSet `  W
)t  ( ( Base `  W
)  X.  ( Base `  W ) ) ) )
5 df-uss 21263 . . . 4  |- UnifSt  =  ( w  e.  _V  |->  ( ( UnifSet `  w )t  (
( Base `  w )  X.  ( Base `  w
) ) ) )
6 ovex 6331 . . . 4  |-  ( (
UnifSet `  W )t  ( (
Base `  W )  X.  ( Base `  W
) ) )  e. 
_V
74, 5, 6fvmpt 5962 . . 3  |-  ( W  e.  _V  ->  (UnifSt `  W )  =  ( ( UnifSet `  W )t  (
( Base `  W )  X.  ( Base `  W
) ) ) )
8 ussval.2 . . . 4  |-  U  =  ( UnifSet `  W )
9 ussval.1 . . . . 5  |-  B  =  ( Base `  W
)
109, 9xpeq12i 4873 . . . 4  |-  ( B  X.  B )  =  ( ( Base `  W
)  X.  ( Base `  W ) )
118, 10oveq12i 6315 . . 3  |-  ( Ut  ( B  X.  B ) )  =  ( (
UnifSet `  W )t  ( (
Base `  W )  X.  ( Base `  W
) ) )
127, 11syl6reqr 2483 . 2  |-  ( W  e.  _V  ->  ( Ut  ( B  X.  B
) )  =  (UnifSt `  W ) )
13 0rest 15321 . . 3  |-  ( (/)t  ( B  X.  B ) )  =  (/)
14 fvprc 5873 . . . . 5  |-  ( -.  W  e.  _V  ->  (
UnifSet `  W )  =  (/) )
158, 14syl5eq 2476 . . . 4  |-  ( -.  W  e.  _V  ->  U  =  (/) )
1615oveq1d 6318 . . 3  |-  ( -.  W  e.  _V  ->  ( Ut  ( B  X.  B
) )  =  (
(/)t 
( B  X.  B
) ) )
17 fvprc 5873 . . 3  |-  ( -.  W  e.  _V  ->  (UnifSt `  W )  =  (/) )
1813, 16, 173eqtr4a 2490 . 2  |-  ( -.  W  e.  _V  ->  ( Ut  ( B  X.  B
) )  =  (UnifSt `  W ) )
1912, 18pm2.61i 168 1  |-  ( Ut  ( B  X.  B ) )  =  (UnifSt `  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1438    e. wcel 1869   _Vcvv 3082   (/)c0 3762    X. cxp 4849   ` cfv 5599  (class class class)co 6303   Basecbs 15114   UnifSetcunif 15193   ↾t crest 15312  UnifStcuss 21260
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-rep 4534  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-reu 2783  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-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  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-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-rest 15314  df-uss 21263
This theorem is referenced by:  ussid  21267  ressuss  21270
  Copyright terms: Public domain W3C validator