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Theorem istps 16690
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 16656 . . 3  |-  TopSp  =  {
f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f
) ) }
21eleq2i 2360 . 2  |-  ( K  e.  TopSp 
<->  K  e.  { f  |  ( TopOpen `  f
)  e.  (TopOn `  ( Base `  f )
) } )
3 topontop 16680 . . . 4  |-  ( J  e.  (TopOn `  A
)  ->  J  e.  Top )
4 0ntop 16667 . . . . . 6  |-  -.  (/)  e.  Top
5 istps.j . . . . . . . 8  |-  J  =  ( TopOpen `  K )
6 fvprc 5535 . . . . . . . 8  |-  ( -.  K  e.  _V  ->  (
TopOpen `  K )  =  (/) )
75, 6syl5eq 2340 . . . . . . 7  |-  ( -.  K  e.  _V  ->  J  =  (/) )
87eleq1d 2362 . . . . . 6  |-  ( -.  K  e.  _V  ->  ( J  e.  Top  <->  (/)  e.  Top ) )
94, 8mtbiri 294 . . . . 5  |-  ( -.  K  e.  _V  ->  -.  J  e.  Top )
109con4i 122 . . . 4  |-  ( J  e.  Top  ->  K  e.  _V )
113, 10syl 15 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  K  e.  _V )
12 fveq2 5541 . . . . 5  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  ( TopOpen `  K )
)
1312, 5syl6eqr 2346 . . . 4  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  J )
14 fveq2 5541 . . . . . 6  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
15 istps.a . . . . . 6  |-  A  =  ( Base `  K
)
1614, 15syl6eqr 2346 . . . . 5  |-  ( f  =  K  ->  ( Base `  f )  =  A )
1716fveq2d 5545 . . . 4  |-  ( f  =  K  ->  (TopOn `  ( Base `  f
) )  =  (TopOn `  A ) )
1813, 17eleq12d 2364 . . 3  |-  ( f  =  K  ->  (
( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) )  <->  J  e.  (TopOn `  A ) ) )
1911, 18elab3 2934 . 2  |-  ( K  e.  { f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) ) }  <-> 
J  e.  (TopOn `  A ) )
202, 19bitri 240 1  |-  ( K  e.  TopSp 
<->  J  e.  (TopOn `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   (/)c0 3468   ` cfv 5271   Basecbs 13164   TopOpenctopn 13342   Topctop 16647  TopOnctopon 16648   TopSpctps 16650
This theorem is referenced by:  istps2  16691  tpspropd  16694  tsettps  16697  indistps2ALT  16767  resstps  16933  prdstps  17339  imastps  17428  xpstopnlem2  17518  tmdtopon  17780  tgptopon  17781  istgp2  17790  oppgtmd  17796  distgp  17798  indistgp  17799  symgtgp  17800  divstgplem  17819  prdstmdd  17822  eltsms  17831  tsmscls  17836  tsmsgsum  17837  tsmsid  17838  tsmsmhm  17844  tsmsadd  17845  dvrcn  17882  cnmpt1vsca  17892  cnmpt2vsca  17893  tlmtgp  17894  isxms2  18010  ressxms  18087  prdsxmslem2  18091  nrgtrg  18216  cnfldtopon  18308  cnmpt1ds  18363  cnmpt2ds  18364  nmcn  18365  cnmpt1ip  18690  cnmpt2ip  18691  csscld  18692  clsocv  18693  minveclem4a  18810  mhmhmeotmd  23315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-top 16652  df-topon 16655  df-topsp 16656
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