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Theorem ussval 19952
Description: The uniform structure on uniform space  W. This proof uses a trick with fvprc 5785 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 5791 . . . . 5  |-  ( w  =  W  ->  ( UnifSet
`  w )  =  ( UnifSet `  W )
)
2 fveq2 5791 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
32, 2xpeq12d 4965 . . . . 5  |-  ( w  =  W  ->  (
( Base `  w )  X.  ( Base `  w
) )  =  ( ( Base `  W
)  X.  ( Base `  W ) ) )
41, 3oveq12d 6210 . . . 4  |-  ( w  =  W  ->  (
( UnifSet `  w )t  (
( Base `  w )  X.  ( Base `  w
) ) )  =  ( ( UnifSet `  W
)t  ( ( Base `  W
)  X.  ( Base `  W ) ) ) )
5 df-uss 19949 . . . 4  |- UnifSt  =  ( w  e.  _V  |->  ( ( UnifSet `  w )t  (
( Base `  w )  X.  ( Base `  w
) ) ) )
6 ovex 6217 . . . 4  |-  ( (
UnifSet `  W )t  ( (
Base `  W )  X.  ( Base `  W
) ) )  e. 
_V
74, 5, 6fvmpt 5875 . . 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 4962 . . . 4  |-  ( B  X.  B )  =  ( ( Base `  W
)  X.  ( Base `  W ) )
118, 10oveq12i 6204 . . 3  |-  ( Ut  ( B  X.  B ) )  =  ( (
UnifSet `  W )t  ( (
Base `  W )  X.  ( Base `  W
) ) )
127, 11syl6reqr 2511 . 2  |-  ( W  e.  _V  ->  ( Ut  ( B  X.  B
) )  =  (UnifSt `  W ) )
13 0rest 14472 . . 3  |-  ( (/)t  ( B  X.  B ) )  =  (/)
14 fvprc 5785 . . . . 5  |-  ( -.  W  e.  _V  ->  (
UnifSet `  W )  =  (/) )
158, 14syl5eq 2504 . . . 4  |-  ( -.  W  e.  _V  ->  U  =  (/) )
1615oveq1d 6207 . . 3  |-  ( -.  W  e.  _V  ->  ( Ut  ( B  X.  B
) )  =  (
(/)t 
( B  X.  B
) ) )
17 fvprc 5785 . . 3  |-  ( -.  W  e.  _V  ->  (UnifSt `  W )  =  (/) )
1813, 16, 173eqtr4a 2518 . 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 1370    e. wcel 1758   _Vcvv 3070   (/)c0 3737    X. cxp 4938   ` cfv 5518  (class class class)co 6192   Basecbs 14278   UnifSetcunif 14352   ↾t crest 14463  UnifStcuss 19946
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 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-rest 14465  df-uss 19949
This theorem is referenced by:  ussid  19953  ressuss  19956
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