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Theorem isusp 20632
Description: The predicate  W is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
Hypotheses
Ref Expression
isusp.1  |-  B  =  ( Base `  W
)
isusp.2  |-  U  =  (UnifSt `  W )
isusp.3  |-  J  =  ( TopOpen `  W )
Assertion
Ref Expression
isusp  |-  ( W  e. UnifSp 
<->  ( U  e.  (UnifOn `  B )  /\  J  =  (unifTop `  U )
) )

Proof of Theorem isusp
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 3127 . 2  |-  ( W  e. UnifSp  ->  W  e.  _V )
2 0nep0 4624 . . . . 5  |-  (/)  =/=  { (/)
}
3 isusp.1 . . . . . . . . . . . 12  |-  B  =  ( Base `  W
)
4 fvprc 5866 . . . . . . . . . . . 12  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
53, 4syl5eq 2520 . . . . . . . . . . 11  |-  ( -.  W  e.  _V  ->  B  =  (/) )
65fveq2d 5876 . . . . . . . . . 10  |-  ( -.  W  e.  _V  ->  (UnifOn `  B )  =  (UnifOn `  (/) ) )
7 ust0 20590 . . . . . . . . . 10  |-  (UnifOn `  (/) )  =  { { (/)
} }
86, 7syl6eq 2524 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (UnifOn `  B )  =  { { (/) } } )
98eleq2d 2537 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  ( U  e.  (UnifOn `  B )  <->  U  e.  { { (/) } } ) )
10 isusp.2 . . . . . . . . . 10  |-  U  =  (UnifSt `  W )
11 fvex 5882 . . . . . . . . . 10  |-  (UnifSt `  W )  e.  _V
1210, 11eqeltri 2551 . . . . . . . . 9  |-  U  e. 
_V
1312elsnc 4057 . . . . . . . 8  |-  ( U  e.  { { (/) } }  <->  U  =  { (/)
} )
149, 13syl6bb 261 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( U  e.  (UnifOn `  B )  <->  U  =  { (/) } ) )
15 fvprc 5866 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (UnifSt `  W )  =  (/) )
1610, 15syl5eq 2520 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  U  =  (/) )
1716eqeq1d 2469 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( U  =  { (/) }  <->  (/)  =  { (/) } ) )
1814, 17bitrd 253 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( U  e.  (UnifOn `  B )  <->  (/)  =  { (/)
} ) )
1918necon3bbid 2714 . . . . 5  |-  ( -.  W  e.  _V  ->  ( -.  U  e.  (UnifOn `  B )  <->  (/)  =/=  { (/)
} ) )
202, 19mpbiri 233 . . . 4  |-  ( -.  W  e.  _V  ->  -.  U  e.  (UnifOn `  B ) )
2120con4i 130 . . 3  |-  ( U  e.  (UnifOn `  B
)  ->  W  e.  _V )
2221adantr 465 . 2  |-  ( ( U  e.  (UnifOn `  B )  /\  J  =  (unifTop `  U )
)  ->  W  e.  _V )
23 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  (UnifSt `  w )  =  (UnifSt `  W ) )
2423, 10syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  (UnifSt `  w )  =  U )
25 fveq2 5872 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
2625, 3syl6eqr 2526 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  B )
2726fveq2d 5876 . . . . 5  |-  ( w  =  W  ->  (UnifOn `  ( Base `  w
) )  =  (UnifOn `  B ) )
2824, 27eleq12d 2549 . . . 4  |-  ( w  =  W  ->  (
(UnifSt `  w )  e.  (UnifOn `  ( Base `  w ) )  <->  U  e.  (UnifOn `  B ) ) )
29 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
30 isusp.3 . . . . . 6  |-  J  =  ( TopOpen `  W )
3129, 30syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  J )
3224fveq2d 5876 . . . . 5  |-  ( w  =  W  ->  (unifTop `  (UnifSt `  w )
)  =  (unifTop `  U
) )
3331, 32eqeq12d 2489 . . . 4  |-  ( w  =  W  ->  (
( TopOpen `  w )  =  (unifTop `  (UnifSt `  w
) )  <->  J  =  (unifTop `  U ) ) )
3428, 33anbi12d 710 . . 3  |-  ( w  =  W  ->  (
( (UnifSt `  w
)  e.  (UnifOn `  ( Base `  w )
)  /\  ( TopOpen `  w )  =  (unifTop `  (UnifSt `  w )
) )  <->  ( U  e.  (UnifOn `  B )  /\  J  =  (unifTop `  U ) ) ) )
35 df-usp 20628 . . 3  |- UnifSp  =  {
w  |  ( (UnifSt `  w )  e.  (UnifOn `  ( Base `  w
) )  /\  ( TopOpen
`  w )  =  (unifTop `  (UnifSt `  w
) ) ) }
3634, 35elab2g 3257 . 2  |-  ( W  e.  _V  ->  ( W  e. UnifSp  <->  ( U  e.  (UnifOn `  B )  /\  J  =  (unifTop `  U ) ) ) )
371, 22, 36pm5.21nii 353 1  |-  ( W  e. UnifSp 
<->  ( U  e.  (UnifOn `  B )  /\  J  =  (unifTop `  U )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118   (/)c0 3790   {csn 4033   ` cfv 5594   Basecbs 14507   TopOpenctopn 14694  UnifOncust 20570  unifTopcutop 20601  UnifStcuss 20624  UnifSpcusp 20625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-res 5017  df-iota 5557  df-fun 5596  df-fv 5602  df-ust 20571  df-usp 20628
This theorem is referenced by:  ressust  20635  ressusp  20636  tususp  20643  uspreg  20645  ucncn  20656  neipcfilu  20667  ucnextcn  20675  xmsuspOLD  20956  xmsusp  20957
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