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Theorem isusp 21354
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 3040 . 2  |-  ( W  e. UnifSp  ->  W  e.  _V )
2 0nep0 4572 . . . . 5  |-  (/)  =/=  { (/)
}
3 isusp.1 . . . . . . . . . . . 12  |-  B  =  ( Base `  W
)
4 fvprc 5873 . . . . . . . . . . . 12  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
53, 4syl5eq 2517 . . . . . . . . . . 11  |-  ( -.  W  e.  _V  ->  B  =  (/) )
65fveq2d 5883 . . . . . . . . . 10  |-  ( -.  W  e.  _V  ->  (UnifOn `  B )  =  (UnifOn `  (/) ) )
7 ust0 21312 . . . . . . . . . 10  |-  (UnifOn `  (/) )  =  { { (/)
} }
86, 7syl6eq 2521 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (UnifOn `  B )  =  { { (/) } } )
98eleq2d 2534 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  ( U  e.  (UnifOn `  B )  <->  U  e.  { { (/) } } ) )
10 isusp.2 . . . . . . . . . 10  |-  U  =  (UnifSt `  W )
11 fvex 5889 . . . . . . . . . 10  |-  (UnifSt `  W )  e.  _V
1210, 11eqeltri 2545 . . . . . . . . 9  |-  U  e. 
_V
1312elsnc 3984 . . . . . . . 8  |-  ( U  e.  { { (/) } }  <->  U  =  { (/)
} )
149, 13syl6bb 269 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( U  e.  (UnifOn `  B )  <->  U  =  { (/) } ) )
15 fvprc 5873 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (UnifSt `  W )  =  (/) )
1610, 15syl5eq 2517 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  U  =  (/) )
1716eqeq1d 2473 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( U  =  { (/) }  <->  (/)  =  { (/) } ) )
1814, 17bitrd 261 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( U  e.  (UnifOn `  B )  <->  (/)  =  { (/)
} ) )
1918necon3bbid 2680 . . . . 5  |-  ( -.  W  e.  _V  ->  ( -.  U  e.  (UnifOn `  B )  <->  (/)  =/=  { (/)
} ) )
202, 19mpbiri 241 . . . 4  |-  ( -.  W  e.  _V  ->  -.  U  e.  (UnifOn `  B ) )
2120con4i 135 . . 3  |-  ( U  e.  (UnifOn `  B
)  ->  W  e.  _V )
2221adantr 472 . 2  |-  ( ( U  e.  (UnifOn `  B )  /\  J  =  (unifTop `  U )
)  ->  W  e.  _V )
23 fveq2 5879 . . . . . 6  |-  ( w  =  W  ->  (UnifSt `  w )  =  (UnifSt `  W ) )
2423, 10syl6eqr 2523 . . . . 5  |-  ( w  =  W  ->  (UnifSt `  w )  =  U )
25 fveq2 5879 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
2625, 3syl6eqr 2523 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  B )
2726fveq2d 5883 . . . . 5  |-  ( w  =  W  ->  (UnifOn `  ( Base `  w
) )  =  (UnifOn `  B ) )
2824, 27eleq12d 2543 . . . 4  |-  ( w  =  W  ->  (
(UnifSt `  w )  e.  (UnifOn `  ( Base `  w ) )  <->  U  e.  (UnifOn `  B ) ) )
29 fveq2 5879 . . . . . 6  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
30 isusp.3 . . . . . 6  |-  J  =  ( TopOpen `  W )
3129, 30syl6eqr 2523 . . . . 5  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  J )
3224fveq2d 5883 . . . . 5  |-  ( w  =  W  ->  (unifTop `  (UnifSt `  w )
)  =  (unifTop `  U
) )
3331, 32eqeq12d 2486 . . . 4  |-  ( w  =  W  ->  (
( TopOpen `  w )  =  (unifTop `  (UnifSt `  w
) )  <->  J  =  (unifTop `  U ) ) )
3428, 33anbi12d 725 . . 3  |-  ( w  =  W  ->  (
( (UnifSt `  w
)  e.  (UnifOn `  ( Base `  w )
)  /\  ( TopOpen `  w )  =  (unifTop `  (UnifSt `  w )
) )  <->  ( U  e.  (UnifOn `  B )  /\  J  =  (unifTop `  U ) ) ) )
35 df-usp 21350 . . 3  |- UnifSp  =  {
w  |  ( (UnifSt `  w )  e.  (UnifOn `  ( Base `  w
) )  /\  ( TopOpen
`  w )  =  (unifTop `  (UnifSt `  w
) ) ) }
3634, 35elab2g 3175 . 2  |-  ( W  e.  _V  ->  ( W  e. UnifSp  <->  ( U  e.  (UnifOn `  B )  /\  J  =  (unifTop `  U ) ) ) )
371, 22, 36pm5.21nii 360 1  |-  ( W  e. UnifSp 
<->  ( U  e.  (UnifOn `  B )  /\  J  =  (unifTop `  U )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031   (/)c0 3722   {csn 3959   ` cfv 5589   Basecbs 15199   TopOpenctopn 15398  UnifOncust 21292  unifTopcutop 21323  UnifStcuss 21346  UnifSpcusp 21347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-res 4851  df-iota 5553  df-fun 5591  df-fv 5597  df-ust 21293  df-usp 21350
This theorem is referenced by:  ressust  21357  ressusp  21358  tususp  21365  uspreg  21367  ucncn  21378  neipcfilu  21389  ucnextcn  21397  xmsusp  21662
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