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Theorem istps 19949
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 19922 . . 3  |-  TopSp  =  {
f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f
) ) }
21eleq2i 2499 . 2  |-  ( K  e.  TopSp 
<->  K  e.  { f  |  ( TopOpen `  f
)  e.  (TopOn `  ( Base `  f )
) } )
3 topontop 19939 . . . 4  |-  ( J  e.  (TopOn `  A
)  ->  J  e.  Top )
4 0ntop 19933 . . . . . 6  |-  -.  (/)  e.  Top
5 istps.j . . . . . . . 8  |-  J  =  ( TopOpen `  K )
6 fvprc 5875 . . . . . . . 8  |-  ( -.  K  e.  _V  ->  (
TopOpen `  K )  =  (/) )
75, 6syl5eq 2475 . . . . . . 7  |-  ( -.  K  e.  _V  ->  J  =  (/) )
87eleq1d 2491 . . . . . 6  |-  ( -.  K  e.  _V  ->  ( J  e.  Top  <->  (/)  e.  Top ) )
94, 8mtbiri 304 . . . . 5  |-  ( -.  K  e.  _V  ->  -.  J  e.  Top )
109con4i 133 . . . 4  |-  ( J  e.  Top  ->  K  e.  _V )
113, 10syl 17 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  K  e.  _V )
12 fveq2 5881 . . . . 5  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  ( TopOpen `  K )
)
1312, 5syl6eqr 2481 . . . 4  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  J )
14 fveq2 5881 . . . . . 6  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
15 istps.a . . . . . 6  |-  A  =  ( Base `  K
)
1614, 15syl6eqr 2481 . . . . 5  |-  ( f  =  K  ->  ( Base `  f )  =  A )
1716fveq2d 5885 . . . 4  |-  ( f  =  K  ->  (TopOn `  ( Base `  f
) )  =  (TopOn `  A ) )
1813, 17eleq12d 2501 . . 3  |-  ( f  =  K  ->  (
( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) )  <->  J  e.  (TopOn `  A ) ) )
1911, 18elab3 3224 . 2  |-  ( K  e.  { f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) ) }  <-> 
J  e.  (TopOn `  A ) )
202, 19bitri 252 1  |-  ( K  e.  TopSp 
<->  J  e.  (TopOn `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    = wceq 1437    e. wcel 1872   {cab 2407   _Vcvv 3080   (/)c0 3761   ` cfv 5601   Basecbs 15120   TopOpenctopn 15319   Topctop 19915  TopOnctopon 19916   TopSpctps 19917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-top 19919  df-topon 19921  df-topsp 19922
This theorem is referenced by:  istps2  19950  tpspropd  19953  tsettps  19956  indistps2ALT  20027  resstps  20201  prdstps  20642  imastps  20734  xpstopnlem2  20824  tmdtopon  21094  tgptopon  21095  istgp2  21104  oppgtmd  21110  distgp  21112  indistgp  21113  symgtgp  21114  qustgplem  21133  prdstmdd  21136  eltsms  21145  tsmscls  21150  tsmsgsum  21151  tsmsid  21152  tsmsmhm  21158  tsmsadd  21159  dvrcn  21196  cnmpt1vsca  21206  cnmpt2vsca  21207  tlmtgp  21208  ressusp  21278  tustps  21286  ucncn  21298  neipcfilu  21309  cnextucn  21316  ucnextcn  21317  isxms2  21461  ressxms  21538  prdsxmslem2  21542  nrgtrg  21690  cnfldtopon  21801  cnmpt1ds  21858  cnmpt2ds  21859  nmcn  21860  cnmpt1ip  22216  cnmpt2ip  22217  csscld  22218  clsocv  22219  minveclem4a  22370  minveclem4aOLD  22382  mhmhmeotmd  28741
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