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Theorem istps 19232
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 19198 . . 3  |-  TopSp  =  {
f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f
) ) }
21eleq2i 2545 . 2  |-  ( K  e.  TopSp 
<->  K  e.  { f  |  ( TopOpen `  f
)  e.  (TopOn `  ( Base `  f )
) } )
3 topontop 19222 . . . 4  |-  ( J  e.  (TopOn `  A
)  ->  J  e.  Top )
4 0ntop 19209 . . . . . 6  |-  -.  (/)  e.  Top
5 istps.j . . . . . . . 8  |-  J  =  ( TopOpen `  K )
6 fvprc 5860 . . . . . . . 8  |-  ( -.  K  e.  _V  ->  (
TopOpen `  K )  =  (/) )
75, 6syl5eq 2520 . . . . . . 7  |-  ( -.  K  e.  _V  ->  J  =  (/) )
87eleq1d 2536 . . . . . 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 5866 . . . . 5  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  ( TopOpen `  K )
)
1312, 5syl6eqr 2526 . . . 4  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  J )
14 fveq2 5866 . . . . . 6  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
15 istps.a . . . . . 6  |-  A  =  ( Base `  K
)
1614, 15syl6eqr 2526 . . . . 5  |-  ( f  =  K  ->  ( Base `  f )  =  A )
1716fveq2d 5870 . . . 4  |-  ( f  =  K  ->  (TopOn `  ( Base `  f
) )  =  (TopOn `  A ) )
1813, 17eleq12d 2549 . . 3  |-  ( f  =  K  ->  (
( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) )  <->  J  e.  (TopOn `  A ) ) )
1911, 18elab3 3257 . 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 1379    e. wcel 1767   {cab 2452   _Vcvv 3113   (/)c0 3785   ` cfv 5588   Basecbs 14490   TopOpenctopn 14677   Topctop 19189  TopOnctopon 19190   TopSpctps 19192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-top 19194  df-topon 19197  df-topsp 19198
This theorem is referenced by:  istps2  19233  tpspropd  19236  tsettps  19239  indistps2ALT  19309  resstps  19482  prdstps  19893  imastps  19985  xpstopnlem2  20075  tmdtopon  20343  tgptopon  20344  istgp2  20353  oppgtmd  20359  distgp  20361  indistgp  20362  symgtgp  20363  divstgplem  20382  prdstmdd  20385  eltsms  20394  tsmscls  20399  tsmsgsum  20400  tsmsid  20401  tsmsgsumOLD  20403  tsmsidOLD  20404  tsmsmhm  20411  tsmsadd  20412  dvrcn  20449  cnmpt1vsca  20459  cnmpt2vsca  20460  tlmtgp  20461  ressusp  20531  tustps  20539  ucncn  20551  neipcfilu  20562  cnextucn  20569  ucnextcn  20570  isxms2  20714  ressxms  20791  prdsxmslem2  20795  nrgtrg  20961  cnfldtopon  21053  cnmpt1ds  21110  cnmpt2ds  21111  nmcn  21112  cnmpt1ip  21450  cnmpt2ip  21451  csscld  21452  clsocv  21453  minveclem4a  21608  mhmhmeotmd  27573
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