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Theorem isusp 19851
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 2996 . 2  |-  ( W  e. UnifSp  ->  W  e.  _V )
2 0nep0 4478 . . . . 5  |-  (/)  =/=  { (/)
}
3 isusp.1 . . . . . . . . . . . 12  |-  B  =  ( Base `  W
)
4 fvprc 5700 . . . . . . . . . . . 12  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
53, 4syl5eq 2487 . . . . . . . . . . 11  |-  ( -.  W  e.  _V  ->  B  =  (/) )
65fveq2d 5710 . . . . . . . . . 10  |-  ( -.  W  e.  _V  ->  (UnifOn `  B )  =  (UnifOn `  (/) ) )
7 ust0 19809 . . . . . . . . . 10  |-  (UnifOn `  (/) )  =  { { (/)
} }
86, 7syl6eq 2491 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (UnifOn `  B )  =  { { (/) } } )
98eleq2d 2510 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  ( U  e.  (UnifOn `  B )  <->  U  e.  { { (/) } } ) )
10 isusp.2 . . . . . . . . . 10  |-  U  =  (UnifSt `  W )
11 fvex 5716 . . . . . . . . . 10  |-  (UnifSt `  W )  e.  _V
1210, 11eqeltri 2513 . . . . . . . . 9  |-  U  e. 
_V
1312elsnc 3916 . . . . . . . 8  |-  ( U  e.  { { (/) } }  <->  U  =  { (/)
} )
149, 13syl6bb 261 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( U  e.  (UnifOn `  B )  <->  U  =  { (/) } ) )
15 fvprc 5700 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (UnifSt `  W )  =  (/) )
1610, 15syl5eq 2487 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  U  =  (/) )
1716eqeq1d 2451 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( U  =  { (/) }  <->  (/)  =  { (/) } ) )
1814, 17bitrd 253 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( U  e.  (UnifOn `  B )  <->  (/)  =  { (/)
} ) )
1918necon3bbid 2657 . . . . 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 5706 . . . . . 6  |-  ( w  =  W  ->  (UnifSt `  w )  =  (UnifSt `  W ) )
2423, 10syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  (UnifSt `  w )  =  U )
25 fveq2 5706 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
2625, 3syl6eqr 2493 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  B )
2726fveq2d 5710 . . . . 5  |-  ( w  =  W  ->  (UnifOn `  ( Base `  w
) )  =  (UnifOn `  B ) )
2824, 27eleq12d 2511 . . . 4  |-  ( w  =  W  ->  (
(UnifSt `  w )  e.  (UnifOn `  ( Base `  w ) )  <->  U  e.  (UnifOn `  B ) ) )
29 fveq2 5706 . . . . . 6  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
30 isusp.3 . . . . . 6  |-  J  =  ( TopOpen `  W )
3129, 30syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  J )
3224fveq2d 5710 . . . . 5  |-  ( w  =  W  ->  (unifTop `  (UnifSt `  w )
)  =  (unifTop `  U
) )
3331, 32eqeq12d 2457 . . . 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 19847 . . 3  |- UnifSp  =  {
w  |  ( (UnifSt `  w )  e.  (UnifOn `  ( Base `  w
) )  /\  ( TopOpen
`  w )  =  (unifTop `  (UnifSt `  w
) ) ) }
3634, 35elab2g 3123 . 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 1369    e. wcel 1756    =/= wne 2620   _Vcvv 2987   (/)c0 3652   {csn 3892   ` cfv 5433   Basecbs 14189   TopOpenctopn 14375  UnifOncust 19789  unifTopcutop 19820  UnifStcuss 19843  UnifSpcusp 19844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-res 4867  df-iota 5396  df-fun 5435  df-fv 5441  df-ust 19790  df-usp 19847
This theorem is referenced by:  ressust  19854  ressusp  19855  tususp  19862  uspreg  19864  ucncn  19875  neipcfilu  19886  ucnextcn  19894  xmsuspOLD  20175  xmsusp  20176
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