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Theorem ussval 20928
Description: The uniform structure on uniform space  W. This proof uses a trick with fvprc 5842 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 5848 . . . . 5  |-  ( w  =  W  ->  ( UnifSet
`  w )  =  ( UnifSet `  W )
)
2 fveq2 5848 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
32sqxpeqd 5014 . . . . 5  |-  ( w  =  W  ->  (
( Base `  w )  X.  ( Base `  w
) )  =  ( ( Base `  W
)  X.  ( Base `  W ) ) )
41, 3oveq12d 6288 . . . 4  |-  ( w  =  W  ->  (
( UnifSet `  w )t  (
( Base `  w )  X.  ( Base `  w
) ) )  =  ( ( UnifSet `  W
)t  ( ( Base `  W
)  X.  ( Base `  W ) ) ) )
5 df-uss 20925 . . . 4  |- UnifSt  =  ( w  e.  _V  |->  ( ( UnifSet `  w )t  (
( Base `  w )  X.  ( Base `  w
) ) ) )
6 ovex 6298 . . . 4  |-  ( (
UnifSet `  W )t  ( (
Base `  W )  X.  ( Base `  W
) ) )  e. 
_V
74, 5, 6fvmpt 5931 . . 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 5010 . . . 4  |-  ( B  X.  B )  =  ( ( Base `  W
)  X.  ( Base `  W ) )
118, 10oveq12i 6282 . . 3  |-  ( Ut  ( B  X.  B ) )  =  ( (
UnifSet `  W )t  ( (
Base `  W )  X.  ( Base `  W
) ) )
127, 11syl6reqr 2514 . 2  |-  ( W  e.  _V  ->  ( Ut  ( B  X.  B
) )  =  (UnifSt `  W ) )
13 0rest 14919 . . 3  |-  ( (/)t  ( B  X.  B ) )  =  (/)
14 fvprc 5842 . . . . 5  |-  ( -.  W  e.  _V  ->  (
UnifSet `  W )  =  (/) )
158, 14syl5eq 2507 . . . 4  |-  ( -.  W  e.  _V  ->  U  =  (/) )
1615oveq1d 6285 . . 3  |-  ( -.  W  e.  _V  ->  ( Ut  ( B  X.  B
) )  =  (
(/)t 
( B  X.  B
) ) )
17 fvprc 5842 . . 3  |-  ( -.  W  e.  _V  ->  (UnifSt `  W )  =  (/) )
1813, 16, 173eqtr4a 2521 . 2  |-  ( -.  W  e.  _V  ->  ( Ut  ( B  X.  B
) )  =  (UnifSt `  W ) )
1912, 18pm2.61i 164 1  |-  ( Ut  ( B  X.  B ) )  =  (UnifSt `  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783    X. cxp 4986   ` cfv 5570  (class class class)co 6270   Basecbs 14716   UnifSetcunif 14794   ↾t crest 14910  UnifStcuss 20922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-rest 14912  df-uss 20925
This theorem is referenced by:  ussid  20929  ressuss  20932
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