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Theorem istps 19415
Description: Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
istps.a  |-  A  =  ( Base `  K
)
istps.j  |-  J  =  ( TopOpen `  K )
Assertion
Ref Expression
istps  |-  ( K  e.  TopSp 
<->  J  e.  (TopOn `  A ) )

Proof of Theorem istps
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 df-topsp 19381 . . 3  |-  TopSp  =  {
f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f
) ) }
21eleq2i 2521 . 2  |-  ( K  e.  TopSp 
<->  K  e.  { f  |  ( TopOpen `  f
)  e.  (TopOn `  ( Base `  f )
) } )
3 topontop 19405 . . . 4  |-  ( J  e.  (TopOn `  A
)  ->  J  e.  Top )
4 0ntop 19392 . . . . . 6  |-  -.  (/)  e.  Top
5 istps.j . . . . . . . 8  |-  J  =  ( TopOpen `  K )
6 fvprc 5850 . . . . . . . 8  |-  ( -.  K  e.  _V  ->  (
TopOpen `  K )  =  (/) )
75, 6syl5eq 2496 . . . . . . 7  |-  ( -.  K  e.  _V  ->  J  =  (/) )
87eleq1d 2512 . . . . . 6  |-  ( -.  K  e.  _V  ->  ( J  e.  Top  <->  (/)  e.  Top ) )
94, 8mtbiri 303 . . . . 5  |-  ( -.  K  e.  _V  ->  -.  J  e.  Top )
109con4i 130 . . . 4  |-  ( J  e.  Top  ->  K  e.  _V )
113, 10syl 16 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  K  e.  _V )
12 fveq2 5856 . . . . 5  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  ( TopOpen `  K )
)
1312, 5syl6eqr 2502 . . . 4  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  J )
14 fveq2 5856 . . . . . 6  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
15 istps.a . . . . . 6  |-  A  =  ( Base `  K
)
1614, 15syl6eqr 2502 . . . . 5  |-  ( f  =  K  ->  ( Base `  f )  =  A )
1716fveq2d 5860 . . . 4  |-  ( f  =  K  ->  (TopOn `  ( Base `  f
) )  =  (TopOn `  A ) )
1813, 17eleq12d 2525 . . 3  |-  ( f  =  K  ->  (
( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) )  <->  J  e.  (TopOn `  A ) ) )
1911, 18elab3 3239 . 2  |-  ( K  e.  { f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) ) }  <-> 
J  e.  (TopOn `  A ) )
202, 19bitri 249 1  |-  ( K  e.  TopSp 
<->  J  e.  (TopOn `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1383    e. wcel 1804   {cab 2428   _Vcvv 3095   (/)c0 3770   ` cfv 5578   Basecbs 14614   TopOpenctopn 14801   Topctop 19372  TopOnctopon 19373   TopSpctps 19375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-top 19377  df-topon 19380  df-topsp 19381
This theorem is referenced by:  istps2  19416  tpspropd  19419  tsettps  19422  indistps2ALT  19493  resstps  19666  prdstps  20108  imastps  20200  xpstopnlem2  20290  tmdtopon  20558  tgptopon  20559  istgp2  20568  oppgtmd  20574  distgp  20576  indistgp  20577  symgtgp  20578  qustgplem  20597  prdstmdd  20600  eltsms  20609  tsmscls  20614  tsmsgsum  20615  tsmsid  20616  tsmsgsumOLD  20618  tsmsidOLD  20619  tsmsmhm  20626  tsmsadd  20627  dvrcn  20664  cnmpt1vsca  20674  cnmpt2vsca  20675  tlmtgp  20676  ressusp  20746  tustps  20754  ucncn  20766  neipcfilu  20777  cnextucn  20784  ucnextcn  20785  isxms2  20929  ressxms  21006  prdsxmslem2  21010  nrgtrg  21176  cnfldtopon  21268  cnmpt1ds  21325  cnmpt2ds  21326  nmcn  21327  cnmpt1ip  21665  cnmpt2ip  21666  csscld  21667  clsocv  21668  minveclem4a  21823  mhmhmeotmd  27887
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