MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isusp Structured version   Unicode version

Theorem isusp 20890
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 3118 . 2  |-  ( W  e. UnifSp  ->  W  e.  _V )
2 0nep0 4627 . . . . 5  |-  (/)  =/=  { (/)
}
3 isusp.1 . . . . . . . . . . . 12  |-  B  =  ( Base `  W
)
4 fvprc 5866 . . . . . . . . . . . 12  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
53, 4syl5eq 2510 . . . . . . . . . . 11  |-  ( -.  W  e.  _V  ->  B  =  (/) )
65fveq2d 5876 . . . . . . . . . 10  |-  ( -.  W  e.  _V  ->  (UnifOn `  B )  =  (UnifOn `  (/) ) )
7 ust0 20848 . . . . . . . . . 10  |-  (UnifOn `  (/) )  =  { { (/)
} }
86, 7syl6eq 2514 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (UnifOn `  B )  =  { { (/) } } )
98eleq2d 2527 . . . . . . . 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 2541 . . . . . . . . 9  |-  U  e. 
_V
1312elsnc 4056 . . . . . . . 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 2510 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  U  =  (/) )
1716eqeq1d 2459 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( U  =  { (/) }  <->  (/)  =  { (/) } ) )
1814, 17bitrd 253 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( U  e.  (UnifOn `  B )  <->  (/)  =  { (/)
} ) )
1918necon3bbid 2704 . . . . 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 2516 . . . . 5  |-  ( w  =  W  ->  (UnifSt `  w )  =  U )
25 fveq2 5872 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
2625, 3syl6eqr 2516 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  B )
2726fveq2d 5876 . . . . 5  |-  ( w  =  W  ->  (UnifOn `  ( Base `  w
) )  =  (UnifOn `  B ) )
2824, 27eleq12d 2539 . . . 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 2516 . . . . 5  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  J )
3224fveq2d 5876 . . . . 5  |-  ( w  =  W  ->  (unifTop `  (UnifSt `  w )
)  =  (unifTop `  U
) )
3331, 32eqeq12d 2479 . . . 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 20886 . . 3  |- UnifSp  =  {
w  |  ( (UnifSt `  w )  e.  (UnifOn `  ( Base `  w
) )  /\  ( TopOpen
`  w )  =  (unifTop `  (UnifSt `  w
) ) ) }
3634, 35elab2g 3248 . 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 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109   (/)c0 3793   {csn 4032   ` cfv 5594   Basecbs 14644   TopOpenctopn 14839  UnifOncust 20828  unifTopcutop 20859  UnifStcuss 20882  UnifSpcusp 20883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fv 5602  df-ust 20829  df-usp 20886
This theorem is referenced by:  ressust  20893  ressusp  20894  tususp  20901  uspreg  20903  ucncn  20914  neipcfilu  20925  ucnextcn  20933  xmsuspOLD  21214  xmsusp  21215
  Copyright terms: Public domain W3C validator