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Theorem istps 12036
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 12035 . . 3  |-  TopSp  =  {
f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f
) ) }
21eleq2i 2179 . 2  |-  ( K  e.  TopSp 
<->  K  e.  { f  |  ( TopOpen `  f
)  e.  (TopOn `  ( Base `  f )
) } )
3 topontop 12018 . . . 4  |-  ( J  e.  (TopOn `  A
)  ->  J  e.  Top )
4 topnfn 11962 . . . . . . 7  |-  TopOpen  Fn  _V
5 fnrel 5177 . . . . . . 7  |-  ( TopOpen  Fn 
_V  ->  Rel  TopOpen )
64, 5ax-mp 7 . . . . . 6  |-  Rel  TopOpen
7 0opn 12010 . . . . . . 7  |-  ( J  e.  Top  ->  (/)  e.  J
)
8 istps.j . . . . . . 7  |-  J  =  ( TopOpen `  K )
97, 8syl6eleq 2205 . . . . . 6  |-  ( J  e.  Top  ->  (/)  e.  (
TopOpen `  K ) )
10 relelfvdm 5405 . . . . . 6  |-  ( ( Rel  TopOpen  /\  (/)  e.  (
TopOpen `  K ) )  ->  K  e.  dom  TopOpen )
116, 9, 10sylancr 408 . . . . 5  |-  ( J  e.  Top  ->  K  e.  dom  TopOpen )
1211elexd 2668 . . . 4  |-  ( J  e.  Top  ->  K  e.  _V )
133, 12syl 14 . . 3  |-  ( J  e.  (TopOn `  A
)  ->  K  e.  _V )
14 fveq2 5373 . . . . 5  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  ( TopOpen `  K )
)
1514, 8syl6eqr 2163 . . . 4  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  J )
16 fveq2 5373 . . . . . 6  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
17 istps.a . . . . . 6  |-  A  =  ( Base `  K
)
1816, 17syl6eqr 2163 . . . . 5  |-  ( f  =  K  ->  ( Base `  f )  =  A )
1918fveq2d 5377 . . . 4  |-  ( f  =  K  ->  (TopOn `  ( Base `  f
) )  =  (TopOn `  A ) )
2015, 19eleq12d 2183 . . 3  |-  ( f  =  K  ->  (
( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) )  <->  J  e.  (TopOn `  A ) ) )
2113, 20elab3 2803 . 2  |-  ( K  e.  { f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) ) }  <-> 
J  e.  (TopOn `  A ) )
222, 21bitri 183 1  |-  ( K  e.  TopSp 
<->  J  e.  (TopOn `  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1312    e. wcel 1461   {cab 2099   _Vcvv 2655   (/)c0 3327   dom cdm 4497   Rel wrel 4502    Fn wfn 5074   ` cfv 5079   Basecbs 11796   TopOpenctopn 11958   Topctop 12001  TopOnctopon 12014   TopSpctps 12034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-cnex 7630  ax-resscn 7631  ax-1re 7633  ax-addrcl 7636
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-inn 8625  df-2 8683  df-3 8684  df-4 8685  df-5 8686  df-6 8687  df-7 8688  df-8 8689  df-9 8690  df-ndx 11799  df-slot 11800  df-base 11802  df-tset 11877  df-rest 11959  df-topn 11960  df-top 12002  df-topon 12015  df-topsp 12035
This theorem is referenced by:  istps2  12037  tpspropd  12040  tsettps  12042  isxms2  12435
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